| L(s) = 1 | − 3·2-s − 2·3-s − 4-s − 15·5-s + 6·6-s + 15·8-s + 2·9-s + 45·10-s − 74·11-s + 2·12-s − 44·13-s + 30·15-s − 61·16-s + 52·17-s − 6·18-s − 168·19-s + 15·20-s + 222·22-s − 124·23-s − 30·24-s + 150·25-s + 132·26-s − 40·27-s + 332·29-s − 90·30-s − 320·31-s + 99·32-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 0.384·3-s − 1/8·4-s − 1.34·5-s + 0.408·6-s + 0.662·8-s + 2/27·9-s + 1.42·10-s − 2.02·11-s + 0.0481·12-s − 0.938·13-s + 0.516·15-s − 0.953·16-s + 0.741·17-s − 0.0785·18-s − 2.02·19-s + 0.167·20-s + 2.15·22-s − 1.12·23-s − 0.255·24-s + 6/5·25-s + 0.995·26-s − 0.285·27-s + 2.12·29-s − 0.547·30-s − 1.85·31-s + 0.546·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14706125 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14706125 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 7 | | \( 1 \) |
| good | 2 | $S_4\times C_2$ | \( 1 + 3 T + 5 p T^{2} + 9 p T^{3} + 5 p^{4} T^{4} + 3 p^{6} T^{5} + p^{9} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 + 2 T + 2 T^{2} + 40 T^{3} + 2 p^{3} T^{4} + 2 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 74 T + 5570 T^{2} + 204680 T^{3} + 5570 p^{3} T^{4} + 74 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 44 T + 3100 T^{2} + 238206 T^{3} + 3100 p^{3} T^{4} + 44 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 52 T + 176 p T^{2} - 454246 T^{3} + 176 p^{4} T^{4} - 52 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 168 T + 26197 T^{2} + 2333344 T^{3} + 26197 p^{3} T^{4} + 168 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 124 T + 38233 T^{2} + 2923048 T^{3} + 38233 p^{3} T^{4} + 124 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 332 T + 80572 T^{2} - 13628846 T^{3} + 80572 p^{3} T^{4} - 332 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 320 T + 113341 T^{2} + 19016064 T^{3} + 113341 p^{3} T^{4} + 320 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 54 T + 139843 T^{2} + 5496260 T^{3} + 139843 p^{3} T^{4} + 54 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 362 T + 248299 T^{2} + 51434996 T^{3} + 248299 p^{3} T^{4} + 362 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 16 T + 149005 T^{2} + 1019664 T^{3} + 149005 p^{3} T^{4} + 16 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 730 T + 427526 T^{2} - 156550492 T^{3} + 427526 p^{3} T^{4} - 730 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 110 T + 105959 T^{2} + 57565396 T^{3} + 105959 p^{3} T^{4} - 110 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 180 T + 3577 T^{2} + 128522760 T^{3} + 3577 p^{3} T^{4} - 180 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 1222 T + 1103759 T^{2} + 593135356 T^{3} + 1103759 p^{3} T^{4} + 1222 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 204 T + 92049 T^{2} + 202232824 T^{3} + 92049 p^{3} T^{4} - 204 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 136 T + 900677 T^{2} + 112926832 T^{3} + 900677 p^{3} T^{4} + 136 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 310 T + 450279 T^{2} + 192471924 T^{3} + 450279 p^{3} T^{4} + 310 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 1034 T + 1094262 T^{2} + 675989052 T^{3} + 1094262 p^{3} T^{4} + 1034 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 20 p T + 2204369 T^{2} - 1855605736 T^{3} + 2204369 p^{3} T^{4} - 20 p^{7} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 242 T + 1427227 T^{2} + 347564516 T^{3} + 1427227 p^{3} T^{4} + 242 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 100 T + 2266056 T^{2} + 184512618 T^{3} + 2266056 p^{3} T^{4} + 100 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−10.60995594645421670388143769536, −10.55799887216616525329177020659, −10.41395946259659566835164556043, −9.990985418542703936746337409647, −9.525361657904853587792311054206, −9.041769494184568811008886938575, −9.032197471101293627578520611794, −8.322667330820568812416062728203, −8.278910882999519101777007002571, −8.019728696666361815861907341690, −7.66284846405910054569810616113, −7.31276585074945345600819315973, −6.90932425211996341018241898328, −6.72071123553706145178269695424, −5.86873427559710920902523044208, −5.81665524176866826002497594441, −5.14343273643395820921210671017, −4.77916952741064063075795841995, −4.60437579747481809019466534505, −3.93618494404550254776334371517, −3.78336931782859209529655010140, −2.88090439599847065294543486071, −2.55146116296370456192564060613, −2.02753531044203201267863712207, −1.12835310643349673566618977629, 0, 0, 0,
1.12835310643349673566618977629, 2.02753531044203201267863712207, 2.55146116296370456192564060613, 2.88090439599847065294543486071, 3.78336931782859209529655010140, 3.93618494404550254776334371517, 4.60437579747481809019466534505, 4.77916952741064063075795841995, 5.14343273643395820921210671017, 5.81665524176866826002497594441, 5.86873427559710920902523044208, 6.72071123553706145178269695424, 6.90932425211996341018241898328, 7.31276585074945345600819315973, 7.66284846405910054569810616113, 8.019728696666361815861907341690, 8.278910882999519101777007002571, 8.322667330820568812416062728203, 9.032197471101293627578520611794, 9.041769494184568811008886938575, 9.525361657904853587792311054206, 9.990985418542703936746337409647, 10.41395946259659566835164556043, 10.55799887216616525329177020659, 10.60995594645421670388143769536
