| L(s) = 1 | + 3·5-s + 25·7-s − 56·11-s − 26·13-s + 13·17-s + 124·19-s − 172·23-s − 79·25-s − 196·29-s + 78·31-s + 75·35-s + 161·37-s − 234·41-s − 135·43-s − 237·47-s − 199·49-s − 666·53-s − 168·55-s + 136·59-s − 146·61-s − 78·65-s + 4·67-s − 563·71-s − 1.48e3·73-s − 1.40e3·77-s + 896·79-s + 1.90e3·83-s + ⋯ |
| L(s) = 1 | + 0.268·5-s + 1.34·7-s − 1.53·11-s − 0.554·13-s + 0.185·17-s + 1.49·19-s − 1.55·23-s − 0.631·25-s − 1.25·29-s + 0.451·31-s + 0.362·35-s + 0.715·37-s − 0.891·41-s − 0.478·43-s − 0.735·47-s − 0.580·49-s − 1.72·53-s − 0.411·55-s + 0.300·59-s − 0.306·61-s − 0.148·65-s + 0.00729·67-s − 0.941·71-s − 2.37·73-s − 2.07·77-s + 1.27·79-s + 2.51·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 3 T + 88 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 25 T + 824 T^{2} - 25 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 56 T + 3154 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 13 T + 3280 T^{2} - 13 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 124 T + 16394 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 172 T + 29102 T^{2} + 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 196 T + 53710 T^{2} + 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 78 T + 40006 T^{2} - 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 161 T + 35352 T^{2} - 161 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 234 T + 40498 T^{2} + 234 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 135 T + 12442 T^{2} + 135 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 237 T + 221232 T^{2} + 237 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 666 T + 406818 T^{2} + 666 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 136 T + 57682 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 146 T + 457466 T^{2} + 146 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 526778 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 563 T + 787016 T^{2} + 563 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1480 T + 1311326 T^{2} + 1480 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 896 T + 1176270 T^{2} - 896 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1902 T + 2047318 T^{2} - 1902 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 272 T + 341902 T^{2} - 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2160 T + 2346718 T^{2} + 2160 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.476366516958382200284847045201, −8.087206963869121225972023405635, −7.81400889849141200233362729284, −7.80396932831637635643053594073, −7.27464702160173075250566866600, −6.76002197705200091642272714307, −6.07819600951342864751100577777, −5.89098509069072670520825994337, −5.23525270373015295894163574208, −5.17570063661038346643307938355, −4.63057170273788552177578929989, −4.34993087928316839902733867080, −3.41301174525124605933704612794, −3.31419484384400358142584548014, −2.54465950286801289453674405269, −2.05741504857799731794972447230, −1.67385491069682334192302001204, −1.13141958712475277380742238075, 0, 0,
1.13141958712475277380742238075, 1.67385491069682334192302001204, 2.05741504857799731794972447230, 2.54465950286801289453674405269, 3.31419484384400358142584548014, 3.41301174525124605933704612794, 4.34993087928316839902733867080, 4.63057170273788552177578929989, 5.17570063661038346643307938355, 5.23525270373015295894163574208, 5.89098509069072670520825994337, 6.07819600951342864751100577777, 6.76002197705200091642272714307, 7.27464702160173075250566866600, 7.80396932831637635643053594073, 7.81400889849141200233362729284, 8.087206963869121225972023405635, 8.476366516958382200284847045201
