| L(s) = 1 | + 2·5-s + 4·7-s − 14·11-s − 24·13-s + 20·17-s − 18·19-s + 46·23-s − 234·25-s + 232·29-s − 292·31-s + 8·35-s − 410·37-s + 300·41-s − 198·43-s + 120·47-s − 674·49-s + 226·53-s − 28·55-s − 24·59-s − 462·61-s − 48·65-s + 190·67-s − 648·71-s − 640·73-s − 56·77-s − 628·79-s − 1.73e3·83-s + ⋯ |
| L(s) = 1 | + 0.178·5-s + 0.215·7-s − 0.383·11-s − 0.512·13-s + 0.285·17-s − 0.217·19-s + 0.417·23-s − 1.87·25-s + 1.48·29-s − 1.69·31-s + 0.0386·35-s − 1.82·37-s + 1.14·41-s − 0.702·43-s + 0.372·47-s − 1.96·49-s + 0.585·53-s − 0.0686·55-s − 0.0529·59-s − 0.969·61-s − 0.0915·65-s + 0.346·67-s − 1.08·71-s − 1.02·73-s − 0.0828·77-s − 0.894·79-s − 2.28·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 685584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 685584 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + 238 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p^{3} T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 14 T + 2074 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 24 T + 3238 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 20 T + 8626 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 18 T - 2126 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8 p T + 55942 T^{2} - 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 292 T + 62126 T^{2} + 292 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 410 T + 117006 T^{2} + 410 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 300 T + 158470 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 198 T + 91738 T^{2} + 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 120 T + 143854 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 226 T + 107398 T^{2} - 226 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 24 T + 141334 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 462 T + 496390 T^{2} + 462 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 190 T + 609498 T^{2} - 190 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 648 T + 71998 T^{2} + 648 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 640 T + 871646 T^{2} + 640 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 628 T + 944066 T^{2} + 628 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1730 T + 1707706 T^{2} + 1730 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 72 T + 1291426 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 760 T + 1823678 T^{2} + 760 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.476712482402802728912968761667, −9.427287924006191622287032872283, −8.633598811882293941472735978370, −8.455221486172850922661535946576, −7.86584918133755598821096188114, −7.52557724220332387314480695696, −7.03555554243476274998302124532, −6.70228953326174016379467800367, −5.89141036455545894008238692322, −5.80159970596869651782703373113, −5.00137580435123011073633756648, −4.93217809030043718903966143156, −3.90449468896079762684953159812, −3.90327258673962952285926172141, −2.80819668361884500830586534419, −2.69347063038882683194647723997, −1.59271428817635787885598493200, −1.49374800440345610498260013793, 0, 0,
1.49374800440345610498260013793, 1.59271428817635787885598493200, 2.69347063038882683194647723997, 2.80819668361884500830586534419, 3.90327258673962952285926172141, 3.90449468896079762684953159812, 4.93217809030043718903966143156, 5.00137580435123011073633756648, 5.80159970596869651782703373113, 5.89141036455545894008238692322, 6.70228953326174016379467800367, 7.03555554243476274998302124532, 7.52557724220332387314480695696, 7.86584918133755598821096188114, 8.455221486172850922661535946576, 8.633598811882293941472735978370, 9.427287924006191622287032872283, 9.476712482402802728912968761667
