| L(s) = 1 | − 3·2-s − 7·4-s − 15·5-s + 35·8-s + 45·10-s + 11-s − 79·13-s − 19·16-s + 72·17-s − 29·19-s + 105·20-s − 3·22-s + 63·23-s + 150·25-s + 237·26-s − 220·29-s + 136·31-s − 73·32-s − 216·34-s + 43·37-s + 87·38-s − 525·40-s + 599·41-s + 170·43-s − 7·44-s − 189·46-s + 3·47-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 7/8·4-s − 1.34·5-s + 1.54·8-s + 1.42·10-s + 0.0274·11-s − 1.68·13-s − 0.296·16-s + 1.02·17-s − 0.350·19-s + 1.17·20-s − 0.0290·22-s + 0.571·23-s + 6/5·25-s + 1.78·26-s − 1.40·29-s + 0.787·31-s − 0.403·32-s − 1.08·34-s + 0.191·37-s + 0.371·38-s − 2.07·40-s + 2.28·41-s + 0.602·43-s − 0.0239·44-s − 0.605·46-s + 0.00931·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 7 | | \( 1 \) |
| good | 2 | $S_4\times C_2$ | \( 1 + 3 T + p^{4} T^{2} + 17 p T^{3} + p^{7} T^{4} + 3 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - T + 3485 T^{2} - 6442 T^{3} + 3485 p^{3} T^{4} - p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 79 T + 6542 T^{2} + 346439 T^{3} + 6542 p^{3} T^{4} + 79 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 72 T + 14635 T^{2} - 688048 T^{3} + 14635 p^{3} T^{4} - 72 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 29 T + 18484 T^{2} + 332301 T^{3} + 18484 p^{3} T^{4} + 29 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 63 T + 32377 T^{2} - 1479946 T^{3} + 32377 p^{3} T^{4} - 63 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 220 T + 74711 T^{2} + 10703064 T^{3} + 74711 p^{3} T^{4} + 220 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 136 T + 27042 T^{2} - 3338174 T^{3} + 27042 p^{3} T^{4} - 136 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 43 T + 33294 T^{2} - 1889771 T^{3} + 33294 p^{3} T^{4} - 43 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 599 T + 272999 T^{2} - 79436618 T^{3} + 272999 p^{3} T^{4} - 599 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 170 T + 62886 T^{2} - 25979064 T^{3} + 62886 p^{3} T^{4} - 170 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 3 T + 72213 T^{2} - 31379318 T^{3} + 72213 p^{3} T^{4} - 3 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 331 T + 442671 T^{2} + 98567438 T^{3} + 442671 p^{3} T^{4} + 331 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 1520 T + 1161665 T^{2} - 599760000 T^{3} + 1161665 p^{3} T^{4} - 1520 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 1160 T + 1094643 T^{2} + 571897312 T^{3} + 1094643 p^{3} T^{4} + 1160 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 806 T + 956790 T^{2} - 441427296 T^{3} + 956790 p^{3} T^{4} - 806 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 406 T + 391273 T^{2} - 58294156 T^{3} + 391273 p^{3} T^{4} - 406 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 1192 T + 1524996 T^{2} + 931572678 T^{3} + 1524996 p^{3} T^{4} + 1192 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 2590 T + 3310582 T^{2} + 2751421632 T^{3} + 3310582 p^{3} T^{4} + 2590 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 508 T + 1655669 T^{2} - 550491384 T^{3} + 1655669 p^{3} T^{4} - 508 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 42 T + 566539 T^{2} - 649223388 T^{3} + 566539 p^{3} T^{4} + 42 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 1020 T + 828679 T^{2} - 17707848 T^{3} + 828679 p^{3} T^{4} + 1020 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−8.225831870938833719537358327650, −7.70126417313012848988247776654, −7.55305459313433055108012034846, −7.46380255277666238963259005029, −7.22372719601776456806403496570, −7.13094383991726968268867384125, −6.54049467637934465317542234818, −6.23388669851040478126330731774, −5.88697566185318166127259745866, −5.79832658248926588059688951324, −5.12540188769477338580568131054, −5.07378539741429688660896089179, −4.92966232592713789277376912834, −4.33585728364535005459578551363, −4.28029753934786921402762023724, −4.17696195862044241446232115207, −3.53069213668194569851421416132, −3.51485445750503009873811372484, −2.96244211844016406723936010424, −2.69528710070449305188675872734, −2.19874225135401749005514921917, −2.12997157977917079782719366324, −1.22294684928163679163864429143, −0.963579792578804111999718284081, −0.887950038680428156048478633861, 0, 0, 0,
0.887950038680428156048478633861, 0.963579792578804111999718284081, 1.22294684928163679163864429143, 2.12997157977917079782719366324, 2.19874225135401749005514921917, 2.69528710070449305188675872734, 2.96244211844016406723936010424, 3.51485445750503009873811372484, 3.53069213668194569851421416132, 4.17696195862044241446232115207, 4.28029753934786921402762023724, 4.33585728364535005459578551363, 4.92966232592713789277376912834, 5.07378539741429688660896089179, 5.12540188769477338580568131054, 5.79832658248926588059688951324, 5.88697566185318166127259745866, 6.23388669851040478126330731774, 6.54049467637934465317542234818, 7.13094383991726968268867384125, 7.22372719601776456806403496570, 7.46380255277666238963259005029, 7.55305459313433055108012034846, 7.70126417313012848988247776654, 8.225831870938833719537358327650
