Primes represented by $x^3-21xy^2+35y^3$.

Not a 'real' answer, but it was too big for a comment. I think that you're looking for a solution without using a calculator or PC but maybe this gives some insight. I did only a quick search with the following bounds: $-50\le x\le50$ and $-50\le y\le50$.

I wrote and ran some Mathematica-code:

In[1]:=Clear["Global`*"];
\[Alpha] = -50;
\[Beta] = 50;
ParallelTable[
  If[TrueQ[PrimeQ[x^3 - 21*x*y^2 + 35*y^3] && 
     x^3 - 21*x*y^2 + 35*y^3 >= 2], {x, y, x^3 - 21*x*y^2 + 35*y^3}, 
   Nothing], {x, \[Alpha], \[Beta]}, {y, \[Alpha], \[Beta]}] //. {} ->
   Nothing

Running the code gives:

Out[1]={{{-48, 25, 1066283}, {-48, 49, 6427331}}, {{-47, -21, 
   7309}, {-47, -15, 127}, {-47, 11, 62189}, {-47, 15, 236377}, {-47, 
   21, 655579}, {-47, 26, 1178549}, {-47, 30, 1729477}}, {{-46, -17, 
   9883}, {-46, -15, 1889}, {-46, 27, 1295783}, {-46, 33, 
   2212433}}, {{-44, -15, 4591}, {-44, 15, 240841}, {-44, 17, 
   353807}, {-44, 23, 829457}, {-44, 35, 2547341}}, {{-43, -20, 
   1693}, {-43, 15, 241793}, {-43, 34, 2340001}, {-43, 40, 
   3605293}, {-43, 45, 4938443}}, {{-41, -18, 5923}, {-41, -15, 
   6679}, {-41, 17, 351863}, {-41, 23, 812393}, {-41, 45, 
   4863979}, {-41, 48, 5785543}}, {{-39, -17, 5417}, {-39, 25, 
   999431}, {-39, 32, 1926217}, {-39, 37, 2834747}, {-39, 43, 
   4237757}}, {{-38, -15, 6553}, {-38, 9, 35281}, {-38, 41, 
   3698801}}, {{-37, -15, 6047}, {-37, 9, 37799}, {-37, 10, 
   62047}, {-37, 16, 291619}, {-37, 21, 616139}, {-37, 39, 
   3207329}, {-37, 40, 3432547}}, {{-36, 7, 2393}, {-36, 13, 
   158003}, {-36, 35, 2380069}, {-36, 37, 2761163}, {-36, 43, 
   4133933}}, {{-34, -15, 3221}, {-34, 7, 7687}, {-34, 27, 
   1170107}, {-34, 37, 2711017}, {-34, 43, 4063627}}, {{-33, -14, 
   3851}, {-33, 14, 195931}, {-33, 16, 284831}, {-33, 26, 
   1047691}, {-33, 34, 2140811}, {-33, 35, 2313613}, {-33, 40, 
   3312863}, {-33, 49, 5745671}}, {{-32, -15, 307}}, {{-31, 10, 
   70309}, {-31, 12, 124433}, {-31, 15, 234809}, {-31, 22, 
   657973}, {-31, 25, 923959}, {-31, 33, 1936943}}, {{-29, -13, 
   1637}, {-29, -10, 1511}, {-29, 8, 32507}, {-29, 12, 123787}, {-29, 
   15, 230761}, {-29, 17, 323567}, {-29, 20, 499211}, {-29, 23, 
   723617}, {-29, 27, 1108477}, {-29, 33, 1896607}, {-29, 38, 
   2775527}, {-29, 45, 4398211}, {-29, 50, 5873111}}, {{-27, -11, 
   2339}, {-27, -10, 2017}, {-27, 29, 1310779}, {-27, 34, 
   2011409}, {-27, 41, 3345679}, {-27, 46, 4586849}, {-27, 50, 
   5772817}}, {{-26, 5, 449}, {-26, 27, 1069363}, {-26, 33, 
   1834813}, {-26, 35, 2151899}, {-26, 47, 4822343}}, {{-24, 7, 
   22877}, {-24, 23, 678637}, {-24, 25, 848051}, {-24, 43, 
   3700817}, {-24, 47, 4733317}}, {{-23, 5, 4283}, {-23, 6, 
   12781}, {-23, 11, 92861}, {-23, 21, 524971}, {-23, 26, 
   929501}, {-23, 29, 1247651}, {-23, 30, 1367533}, {-23, 39, 
   2798641}, {-23, 50, 5570333}}, {{-22, -9, 1259}, {-22, 9, 
   52289}, {-22, 15, 211427}, {-22, 19, 396199}, {-22, 21, 
   517229}, {-22, 25, 824977}, {-22, 45, 4114277}}, {{-19, -8, 
   757}, {-19, 7, 24697}, {-19, 10, 68041}, {-19, 18, 326537}, {-19, 
   22, 558937}, {-19, 25, 789391}, {-19, 28, 1074277}, {-19, 33, 
   1685447}, {-19, 42, 3290057}, {-19, 43, 3513637}, {-19, 48, 
   4783157}}, {{-18, 5, 7993}, {-18, 11, 86491}, {-18, 41, 
   3041821}}, {{-17, -6, 379}, {-17, 5, 8387}, {-17, 11, 84869}, {-17,
    21, 476659}, {-17, 24, 684559}, {-17, 30, 1261387}, {-17, 35, 
   1933037}, {-17, 36, 2090719}, {-17, 44, 3667679}}, {{-16, 7, 
   24373}, {-16, 33, 1619603}}, {{-13, -6, 71}, {-13, 10, 
   60103}, {-13, 16, 211051}, {-13, 25, 715303}, {-13, 31, 
   1302841}, {-13, 34, 1689031}, {-13, 36, 1984571}}, {{-12, -5, 
   197}, {-12, 19, 329309}, {-12, 31, 1283129}}, {{-11, 3, 
   1693}, {-11, 5, 8819}, {-11, 12, 92413}, {-11, 15, 168769}, {-11, 
   20, 371069}, {-11, 30, 1151569}, {-11, 35, 1782269}, {-11, 38, 
   2252753}, {-11, 42, 2999233}, {-11, 47, 4142753}}, {{-9, 2, 
   307}, {-9, 8, 29287}, {-9, 10, 53171}, {-9, 13, 108107}, {-9, 25, 
   664271}, {-9, 32, 1339687}, {-9, 35, 1731421}, {-9, 43, 
   3131477}, {-9, 50, 4846771}}, {{-8, 9, 38611}, {-8, 15, 
   155413}, {-8, 29, 994391}, {-8, 45, 3529063}}, {{-6, 5, 7309}, {-6,
    13, 97973}, {-6, 25, 625409}, {-6, 43, 3015503}, {-6, 47, 
   3911923}}, {{-4, 3, 1637}, {-4, 7, 16057}, {-4, 27, 750077}, {-4, 
   33, 1349207}}, {{-3, 1, 71}, {-3, 4, 3221}, {-3, 5, 5923}, {-3, 11,
    54181}, {-3, 19, 262781}, {-3, 40, 2340773}, {-3, 44, 
   3103381}, {-3, 46, 3540041}, {-3, 49, 4268951}}, {{-2, 5, 
   5417}, {-2, 9, 28909}, {-2, 11, 51659}}, {{-1, 7, 13033}, {-1, 15, 
   122849}, {-1, 18, 210923}, {-1, 22, 382843}, {-1, 27, 704213}, {-1,
    30, 963899}, {-1, 40, 2273599}, {-1, 43, 2821573}}, {{1, 2, 
   197}, {1, 3, 757}, {1, 5, 3851}, {1, 12, 57457}, {1, 17, 
   165887}, {1, 23, 414737}, {1, 35, 1474901}}, {{2, 19, 224911}, {2, 
   21, 305621}, {2, 25, 520633}}, {{3, 4, 1259}, {3, 14, 83719}, {3, 
   20, 254827}, {3, 26, 572599}, {3, 29, 800659}, {3, 34, 
   1302839}, {3, 40, 2139227}, {3, 44, 2859499}}, {{4, 5, 2339}, {4, 
   15, 99289}, {4, 17, 147743}, {4, 27, 627733}, {4, 33, 1166383}, {4,
    45, 3019339}}, {{6, 7, 6047}, {6, 13, 55817}, {6, 17, 135757}, {6,
    23, 359407}, {6, 35, 1346491}}, {{8, 1, 379}, {8, 45, 
   2849687}, {8, 49, 3714859}}, {{9, 5, 379}, {9, 8, 6553}, {9, 10, 
   16829}, {9, 20, 205129}, {9, 22, 281933}, {9, 23, 326593}, {9, 43, 
   2434013}}, {{11, -2, 127}, {11, 3, 197}, {11, 7, 2017}, {11, 12, 
   28547}, {11, 15, 67481}, {11, 25, 403831}, {11, 45, 
   2722931}}, {{12, 1, 1511}, {12, 25, 391103}, {12, 35, 
   1193653}, {12, 49, 3514391}}, {{13, -1, 1889}, {13, 11, 
   15749}, {13, 14, 44729}, {13, 15, 58897}, {13, 24, 328789}, {13, 
   30, 701497}, {13, 35, 1168397}, {13, 36, 1281349}, {13, 45, 
   2638747}}, {{16, -3, 127}, {16, 3, 2017}, {16, 5, 71}, {16, 27, 
   448057}, {16, 33, 895987}}, {{17, 1, 4591}, {17, 9, 1511}, {17, 19,
    116101}, {17, 24, 283121}, {17, 31, 704521}, {17, 39, 
   1538081}, {17, 40, 1673713}, {17, 46, 2656261}}, {{18, -1, 
   5419}, {18, 5, 757}, {18, 11, 6679}, {18, 29, 541549}}, {{19, 5, 
   1259}, {19, 12, 9883}, {19, 18, 81703}, {19, 30, 592759}, {19, 33, 
   830143}, {19, 35, 1018709}, {19, 45, 2388259}}, {{22, -1, 
   10151}, {22, 21, 131041}, {22, 29, 475721}, {22, 41, 
   1646261}}, {{23, 1, 11719}, {23, 4, 6679}, {23, 6, 2339}, {23, 15, 
   21617}, {23, 39, 1353689}, {23, 45, 2223467}}, {{24, 17, 
   40123}, {24, 23, 173053}, {24, 35, 897049}, {24, 37, 
   1096703}}, {{26, -3, 11717}, {26, 27, 308447}}, {{27, -1, 
   19081}, {27, 5, 9883}, {27, 14, 4591}, {27, 16, 17891}, {27, 19, 
   55061}, {27, 20, 72883}, {27, 25, 212183}, {27, 31, 517481}, {27, 
   35, 825733}}, {{29, -5, 4789}, {29, -2, 21673}, {29, 3, 
   19853}, {29, 7, 6553}, {29, 18, 31193}, {29, 25, 190639}, {29, 27, 
   269333}}, {{31, 3, 24877}, {31, 5, 17891}, {31, 8, 6047}, {31, 20, 
   49391}, {31, 30, 388891}, {31, 32, 510047}, {31, 33, 578647}, {31, 
   45, 1900891}}, {{32, -5, 11593}, {32, 9, 3851}, {32, 19, 
   30241}, {32, 31, 429661}}, {{33, 1, 35279}, {33, 10, 1637}, {33, 
   16, 1889}, {33, 20, 38737}, {33, 29, 306739}, {33, 34, 
   610469}, {33, 35, 687637}, {33, 46, 1976309}, {33, 49, 
   2489759}, {33, 50, 2678437}}, {{34, 23, 87443}, {34, 33, 
   519553}, {34, 35, 665279}, {34, 45, 1782829}}, {{36, 7, 
   21617}, {36, 17, 127}, {36, 23, 72577}, {36, 37, 784547}, {36, 43, 
   1431557}}, {{37, -6, 15121}, {37, 5, 35603}, {37, 6, 30241}, {37, 
   11, 3221}, {37, 20, 19853}, {37, 30, 296353}, {37, 41, 
   1156751}}, {{38, 9, 15749}, {38, 31, 330679}}, {{39, -5, 
   34469}, {39, -2, 55763}, {39, 7, 31193}, {39, 20, 11719}, {39, 22, 
   35603}, {39, 23, 51913}, {39, 28, 185543}}, {{41, 7, 38737}, {41, 
   12, 5417}, {41, 13, 307}, {41, 22, 24877}, {41, 43, 
   1259677}}, {{43, -6, 39439}, {43, -4, 62819}, {43, -1, 78569}, {43,
    6, 54559}, {43, 11, 16829}, {43, 21, 5419}, {43, 26, 84239}, {43, 
   29, 173699}, {43, 39, 782209}, {43, 44, 1312739}}, {{44, -5, 
   57709}, {44, 3, 77813}, {44, 7, 51913}, {44, 13, 5923}, {44, 25, 
   54559}, {44, 27, 100493}, {44, 37, 593083}, {44, 45, 
   1403459}}, {{46, -7, 37997}, {46, -3, 87697}, {46, 33, 
   303157}, {46, 35, 414611}}, {{47, 1, 102871}, {47, 4, 90271}, {47, 
   9, 49391}, {47, 10, 40123}, {47, 39, 678761}, {47, 40, 
   764623}}, {{48, -5, 81017}, {48, 1, 109619}, {48, 5, 89767}, {48, 
   35, 376417}, {48, 41, 828379}}}

So, with the bounds $-50\le x\le50$ and $-50\le y\le50$ we find $402$ solutions. In order to find that I used:

In[2]:=Clear["Global`*"];
\[Alpha] = -50;
\[Beta] = 50;
f = Total@*Map[Length];
f[ParallelTable[
   If[TrueQ[
     PrimeQ[x^3 - 21*x*y^2 + 35*y^3] && 
      x^3 - 21*x*y^2 + 35*y^3 >= 2], {x, y, x^3 - 21*x*y^2 + 35*y^3}, 
    Nothing], {x, \[Alpha], \[Beta]}, {y, \[Alpha], \[Beta]}] //. {} \
-> Nothing]

Out[2]=402

If we extend the bounds to $-10^3\le x\le10^3$ and $-10^3\le y\le10^3$ we find $92522$ solutions. If we extend the bounds, again, to $-10^4\le x\le10^4$ and $-10^4\le y\le10^4$ we find $6950603$ solutions.

Let $\alpha$ be a root of $x^3-21x+35=0$. Then it is easy to characterize primes of form $$N(x + y\alpha + z\alpha^2) = x^3+35y^3+1225z^3-105xyz-21xy^2+441xz^2+42x^2z-735yz^2$$ which was already eluded in Will Jagy's answer.

(Theorem) A prime $p\neq 3,7$ can be represented by above cubic form iff $p\equiv \pm 1, \pm 8 \pmod{63}$.

Proof of Theorem: Let $K$ be number field of $x^3-21x+35$. I assume the following facts: $K$ has class number $3$, contained in $\mathbb{Q}(\zeta_{63})$.

Let $H$ be the Hilbert class field of $K$, then $H/\mathbb{Q}$ is abelian of degree $9$ ($H/\mathbb{Q}$ is Galois and any group of order $9$ is abelian).

1. Claim: $H\subset \mathbb{Q}(\zeta_{63})$. This follows from a general (but not well-known) fact of cyclotomic fields. We have the following proposition, proved in the answer here: If $F/\mathbb{Q}(\zeta_m)$ is unramified (at finite primes) and $F/\mathbb{Q}$ abelian, then $F=\mathbb{Q}(\zeta_m)$. Because $H/\mathbb{Q}$ is abelian, applying this proposition to $F=H\mathbb{Q}(\zeta_{63})$ shows that $H\mathbb{Q}(\zeta_{63}) = \mathbb{Q}(\zeta_{63})$, so $H\subset \mathbb{Q}(\zeta_{63})$.
2. Claim: $H$ corresponds to $\{\pm 1,\pm 8\} \subset (\mathbb{Z}/63\mathbb{Z})^\times$. $H$ corresponds to a subgroup of order $4$ of $(\mathbb{Z}/63\mathbb{Z})^\times = C_6 \times C_6$, such a subgroup is unique, and this is the only one.

Finally $p\neq 3,7$ can be represented as $N(x + y\alpha + z\alpha^2)$ iff $p$ splits into principle ideals in $K$, iff $p$ splits completely in $H$, completing the proof.

---

Restricting to $z=0$ of the cubic form is more tricky, and likely has no simple answer. If $\pi(n)$ denotes the prime counting function, then

The equation of form $N(x+y\alpha)$ is a Thue equation. For each individual $p$, there is an algorithm to check whether $N(x+y\alpha) = p$ has integral solution. The following Magma code checks above table for small $p$:

R<x> := PolynomialRing(Integers());
f := x^3 -21*x+35;
T := Thue(f);
list := {71, 127, 181, 197, 251, 307, 379, 433, 449, 503, 631, 701, 757, 811};
t := { n : n in list | Solutions(T, n) ne [] };
t

which outputs { 71, 127, 197, 307, 379, 449, 757 }. The complete list of primes $p$ with $\pi(p)\leq 9000$ which can be written as $p=x^3-21xy^2+35y^3$ is

{71,127,197,307,379,449,757,827,1259,1511,1637,1693,1889,2017,2339,2393,3221,3851,4283,4591,4789,5417,5419,5923,6047,6229,6553,6679,6733,7127,7253,7309,7687,7993,8387,8819,9883,10151,11593,11717,11719,12781,13033,14057,14923,15121,15749,16057,16829,17891,19081,19853,20593,21617,21673,22877,23633,24373,24697,24877,26641,28351,28547,28909,29287,30241,30493,31193,32381,32507,34469,35279,35281,35603,37799,37997,38611,38737,39439,40123,41887,42013,42407,44281,44729,45863,46187,47431,47881,49391,51659,51913,52289,53171,53857,54181,54559,55061,55763,55817,57457,57709,58897,60103,61487,62047,62189,62819,66403,67481,68041,70309,72269,72577,72883,77813,78569,79813,81017,81019,81703,82727,83719,84239,84869,86491,87443,87697,89767,90019,90271,92177,92357,92413,92861}

The discriminant of $x^3 - 21 x + 35$ is a square, many things drop out. The primes represented by the full norm form you give will be primes that are $$ 1, 5, 8, 11, 23, 25, \pmod{63} $$ $$ 62, 58, 55, 52, 40, 38, \pmod{63} $$

There is more restriction, not obvious initially, it is a subgroup of the residues $$ \color{red}{ 1, 8, 55, 62 \pmod{63} } $$ $$x^3+35y^3+1225z^3-105xyz-21xy^2+441xz^2+42x^2z-735yz^2.$$

What restrictions we get with $z=0$ are anyone's guess.

Note that $x^3 - 21 x + 35$ and $x^3 - 21 x + 28$ give different fields