| L(s) = 1 | + 4·2-s − 6·3-s + 12·4-s + 5·5-s − 24·6-s − 14·7-s + 32·8-s + 27·9-s + 20·10-s − 32·11-s − 72·12-s − 26·13-s − 56·14-s − 30·15-s + 80·16-s − 78·17-s + 108·18-s − 9·19-s + 60·20-s + 84·21-s − 128·22-s + 17·23-s − 192·24-s − 205·25-s − 104·26-s − 108·27-s − 168·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.447·5-s − 1.63·6-s − 0.755·7-s + 1.41·8-s + 9-s + 0.632·10-s − 0.877·11-s − 1.73·12-s − 0.554·13-s − 1.06·14-s − 0.516·15-s + 5/4·16-s − 1.11·17-s + 1.41·18-s − 0.108·19-s + 0.670·20-s + 0.872·21-s − 1.24·22-s + 0.154·23-s − 1.63·24-s − 1.63·25-s − 0.784·26-s − 0.769·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - p T + 46 p T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 32 T + 2498 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 78 T + 8722 T^{2} + 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 9 T + 9302 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 17 T + 19970 T^{2} - 17 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 67 T + 27824 T^{2} + 67 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 121 T + 14706 T^{2} + 121 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 30 T + 71186 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 570 T + 201322 T^{2} + 570 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + p T + 153570 T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 293 T + 137732 T^{2} + 293 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 555 T + 193924 T^{2} + 555 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 562 T + 488774 T^{2} + 562 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1060 T + 719742 T^{2} + 1060 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 130 T + 373806 T^{2} + 130 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2 T - 486322 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1379 T + 1252788 T^{2} + 1379 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 913 T + 926694 T^{2} - 913 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 363 T + 178360 T^{2} - 363 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 315 T + 836218 T^{2} - 315 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 599 T + 1363140 T^{2} + 599 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.30536951635691210989071048771, −10.05077144647553301172166080050, −9.283496287308514536436207526407, −9.201747600080390262344900604329, −8.050205721701552824871896088791, −7.83680143586349202506360944358, −7.13508038206158898621610989809, −6.80001894931559616168808531832, −6.25872739277498976758114174734, −6.04987416309986973573535021655, −5.35658726040189688495352687733, −5.20157917498516533829207657861, −4.48353441673830814066918005853, −4.20574355703643239077945068406, −3.27226146622143242476150069206, −2.98204653913667665445108772460, −1.89716309205247239044594928329, −1.74458790150880540260686909809, 0, 0,
1.74458790150880540260686909809, 1.89716309205247239044594928329, 2.98204653913667665445108772460, 3.27226146622143242476150069206, 4.20574355703643239077945068406, 4.48353441673830814066918005853, 5.20157917498516533829207657861, 5.35658726040189688495352687733, 6.04987416309986973573535021655, 6.25872739277498976758114174734, 6.80001894931559616168808531832, 7.13508038206158898621610989809, 7.83680143586349202506360944358, 8.050205721701552824871896088791, 9.201747600080390262344900604329, 9.283496287308514536436207526407, 10.05077144647553301172166080050, 10.30536951635691210989071048771
