| L(s) = 1 | + 11·5-s − 7-s + 116·11-s − 26·13-s − 51·17-s − 128·19-s − 196·23-s − 79·25-s + 12·29-s − 226·31-s − 11·35-s − 143·37-s + 278·41-s − 253·43-s − 867·47-s + 37·49-s − 666·53-s + 1.27e3·55-s + 996·59-s − 66·61-s − 286·65-s − 1.56e3·67-s + 923·71-s − 424·73-s − 116·77-s + 408·79-s − 1.05e3·83-s + ⋯ |
| L(s) = 1 | + 0.983·5-s − 0.0539·7-s + 3.17·11-s − 0.554·13-s − 0.727·17-s − 1.54·19-s − 1.77·23-s − 0.631·25-s + 0.0768·29-s − 1.30·31-s − 0.0531·35-s − 0.635·37-s + 1.05·41-s − 0.897·43-s − 2.69·47-s + 0.107·49-s − 1.72·53-s + 3.12·55-s + 2.19·59-s − 0.138·61-s − 0.545·65-s − 2.84·67-s + 1.54·71-s − 0.679·73-s − 0.171·77-s + 0.581·79-s − 1.39·83-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 | $D_{4}$ | \( 1 - 11 T + 8 p^{2} T^{2} - 11 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T - 36 T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 58 T + p^{3} T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 3 p T + 6544 T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 128 T + 870 p T^{2} + 128 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 196 T + 22382 T^{2} + 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 226 T + 64326 T^{2} + 226 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 143 T + 92856 T^{2} + 143 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 278 T + 118322 T^{2} - 278 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 253 T + 65154 T^{2} + 253 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 867 T + 391636 T^{2} + 867 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 666 T + 336418 T^{2} + 666 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 996 T + 576586 T^{2} - 996 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 66 T + 254426 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1560 T + 1208642 T^{2} + 1560 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 13 p T + 751532 T^{2} - 13 p^{4} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 424 T + 811422 T^{2} + 424 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 408 T + 945518 T^{2} - 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1058 T + 1330646 T^{2} + 1058 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1376 T + 1871726 T^{2} - 1376 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 592 T + 1901406 T^{2} - 592 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.831258463647327453676047222812, −8.458960202447343828045228549678, −7.85556768861948127372046589260, −7.55281205206702326125417887914, −6.79541293239646521201810552273, −6.61511948423511529007429779560, −6.28419156042322636267208785645, −6.14625106531040608197397669566, −5.57700461525354624120028896574, −4.97881797899488226897228068290, −4.41395251772722138303326954474, −4.17083155835992394410733638451, −3.63540359289829990501531118571, −3.44482134323612929253075656835, −2.31760471267886097128858749520, −2.11566073900563348503701162433, −1.47203572202887599746169748178, −1.41293365167658826768440586525, 0, 0,
1.41293365167658826768440586525, 1.47203572202887599746169748178, 2.11566073900563348503701162433, 2.31760471267886097128858749520, 3.44482134323612929253075656835, 3.63540359289829990501531118571, 4.17083155835992394410733638451, 4.41395251772722138303326954474, 4.97881797899488226897228068290, 5.57700461525354624120028896574, 6.14625106531040608197397669566, 6.28419156042322636267208785645, 6.61511948423511529007429779560, 6.79541293239646521201810552273, 7.55281205206702326125417887914, 7.85556768861948127372046589260, 8.458960202447343828045228549678, 8.831258463647327453676047222812
