| L(s) = 1 | + 6·2-s − 9·3-s + 24·4-s − 5-s − 54·6-s + 21·7-s + 80·8-s + 54·9-s − 6·10-s + 63·11-s − 216·12-s − 39·13-s + 126·14-s + 9·15-s + 240·16-s − 17-s + 324·18-s − 39·19-s − 24·20-s − 189·21-s + 378·22-s + 143·23-s − 720·24-s − 156·25-s − 234·26-s − 270·27-s + 504·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1.73·3-s + 3·4-s − 0.0894·5-s − 3.67·6-s + 1.13·7-s + 3.53·8-s + 2·9-s − 0.189·10-s + 1.72·11-s − 5.19·12-s − 0.832·13-s + 2.40·14-s + 0.154·15-s + 15/4·16-s − 0.0142·17-s + 4.24·18-s − 0.470·19-s − 0.268·20-s − 1.96·21-s + 3.66·22-s + 1.29·23-s − 6.12·24-s − 1.24·25-s − 1.76·26-s − 1.92·27-s + 3.40·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(19.27728323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(19.27728323\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 3 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 7 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 13 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 + T + 157 T^{2} - 206 T^{3} + 157 p^{3} T^{4} + p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 63 T + 3351 T^{2} - 146050 T^{3} + 3351 p^{3} T^{4} - 63 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + T + 363 T^{2} - 623450 T^{3} + 363 p^{3} T^{4} + p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 39 T + 5805 T^{2} + 699994 T^{3} + 5805 p^{3} T^{4} + 39 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 143 T + 25389 T^{2} - 2850434 T^{3} + 25389 p^{3} T^{4} - 143 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 307 T + 86655 T^{2} - 14094874 T^{3} + 86655 p^{3} T^{4} - 307 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 12 T - 10467 T^{2} + 2616344 T^{3} - 10467 p^{3} T^{4} - 12 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 429 T + 177243 T^{2} - 39377710 T^{3} + 177243 p^{3} T^{4} - 429 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 210 T + 170379 T^{2} - 29770788 T^{3} + 170379 p^{3} T^{4} - 210 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 689 T + 351121 T^{2} - 107955862 T^{3} + 351121 p^{3} T^{4} - 689 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 582 T + 399405 T^{2} - 123673268 T^{3} + 399405 p^{3} T^{4} - 582 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 474 T + 421635 T^{2} - 126365564 T^{3} + 421635 p^{3} T^{4} - 474 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 74 T + 585769 T^{2} + 27500348 T^{3} + 585769 p^{3} T^{4} + 74 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 1045 T + 1030575 T^{2} - 511019462 T^{3} + 1030575 p^{3} T^{4} - 1045 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 826 T + 667721 T^{2} - 353225980 T^{3} + 667721 p^{3} T^{4} - 826 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 630 T + 1181061 T^{2} - 455351220 T^{3} + 1181061 p^{3} T^{4} - 630 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 407 T + 867899 T^{2} - 352120634 T^{3} + 867899 p^{3} T^{4} - 407 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 942 T + 1257357 T^{2} - 933087076 T^{3} + 1257357 p^{3} T^{4} - 942 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 148 T + 1127909 T^{2} + 112520280 T^{3} + 1127909 p^{3} T^{4} + 148 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 940 T + 1295487 T^{2} + 731288344 T^{3} + 1295487 p^{3} T^{4} + 940 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 46 T + 1729151 T^{2} - 188998372 T^{3} + 1729151 p^{3} T^{4} - 46 p^{6} T^{5} + p^{9} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.496866533334924849831970208835, −8.935236342766734839247363085125, −8.395323291167980654762660740154, −8.365955692273927008744446953824, −7.64760772654214340912352262501, −7.45402868384674430288403239055, −7.26900158835149054914061387053, −6.86794663428089351509660255608, −6.47202119438357719943790460615, −6.44515463518534440328089629070, −5.81532852259295466951472036945, −5.80715831695595891951541742014, −5.49836972508644200486616307014, −4.86092429990001942612819504292, −4.74171215150132614890961168052, −4.66134100222113347350738363389, −3.91626105830319781734631142479, −3.86062720984880498343155847378, −3.84492296624682310538012646114, −2.60698312010280308631417270104, −2.45268386148788617400712178778, −2.22535777456749025379456377694, −1.25803290802505524979671684274, −0.967265082058800699404207436367, −0.76757527469843736963342837434,
0.76757527469843736963342837434, 0.967265082058800699404207436367, 1.25803290802505524979671684274, 2.22535777456749025379456377694, 2.45268386148788617400712178778, 2.60698312010280308631417270104, 3.84492296624682310538012646114, 3.86062720984880498343155847378, 3.91626105830319781734631142479, 4.66134100222113347350738363389, 4.74171215150132614890961168052, 4.86092429990001942612819504292, 5.49836972508644200486616307014, 5.80715831695595891951541742014, 5.81532852259295466951472036945, 6.44515463518534440328089629070, 6.47202119438357719943790460615, 6.86794663428089351509660255608, 7.26900158835149054914061387053, 7.45402868384674430288403239055, 7.64760772654214340912352262501, 8.365955692273927008744446953824, 8.395323291167980654762660740154, 8.935236342766734839247363085125, 9.496866533334924849831970208835
