| L(s) = 1 | − 2·2-s − 3·3-s − 5·5-s + 6·6-s + 26·7-s + 8·8-s + 10·10-s − 36·11-s − 21·13-s − 52·14-s + 15·15-s − 16·16-s + 32·17-s + 152·19-s − 78·21-s + 72·22-s + 218·23-s − 24·24-s + 42·26-s + 27·27-s − 190·29-s − 30·30-s + 182·31-s + 108·33-s − 64·34-s − 130·35-s + 230·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 0.447·5-s + 0.408·6-s + 1.40·7-s + 0.353·8-s + 0.316·10-s − 0.986·11-s − 0.448·13-s − 0.992·14-s + 0.258·15-s − 1/4·16-s + 0.456·17-s + 1.83·19-s − 0.810·21-s + 0.697·22-s + 1.97·23-s − 0.204·24-s + 0.316·26-s + 0.192·27-s − 1.21·29-s − 0.182·30-s + 1.05·31-s + 0.569·33-s − 0.322·34-s − 0.627·35-s + 1.02·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.861753900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.861753900\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 19 | $C_2$ | \( 1 - 8 p T + p^{3} T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 13 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 18 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 21 T - 1756 T^{2} + 21 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 32 T - 3889 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 218 T + 35357 T^{2} - 218 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 190 T + 11711 T^{2} + 190 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 91 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 115 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 378 T + 73963 T^{2} - 378 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 335 T + 32718 T^{2} - 335 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 300 T - 13823 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 496 T + 97139 T^{2} - 496 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 582 T + 133345 T^{2} + 582 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 875 T + 538644 T^{2} - 875 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 105 T - 289738 T^{2} + 105 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 680 T + 104489 T^{2} + 680 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 157 T - 364368 T^{2} - 157 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 71 T - 487998 T^{2} + 71 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 914 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 1080 T + 461431 T^{2} - 1080 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 754 T - 344157 T^{2} - 754 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.51161540605892350177548527885, −10.23339827904877323287404567572, −9.630910632572267685277041297810, −9.212043165415439324134721776624, −8.868780321143200362504093733595, −8.270976930398169323765385742278, −7.68973757672204192979212413031, −7.57279208759656659075564344773, −7.37070350115969873481850763540, −6.59625503353062488325329772245, −5.79442624454068534756735316389, −5.25962649065079262064578399074, −5.24302579275611335261727156661, −4.48530601916834858786012972945, −4.11080917958857012725495887257, −3.02496748295000358603869642234, −2.74080562955132797625437455789, −1.75727842967995925301706414328, −0.908328759149631836261437569197, −0.66757323209297836823374018417,
0.66757323209297836823374018417, 0.908328759149631836261437569197, 1.75727842967995925301706414328, 2.74080562955132797625437455789, 3.02496748295000358603869642234, 4.11080917958857012725495887257, 4.48530601916834858786012972945, 5.24302579275611335261727156661, 5.25962649065079262064578399074, 5.79442624454068534756735316389, 6.59625503353062488325329772245, 7.37070350115969873481850763540, 7.57279208759656659075564344773, 7.68973757672204192979212413031, 8.270976930398169323765385742278, 8.868780321143200362504093733595, 9.212043165415439324134721776624, 9.630910632572267685277041297810, 10.23339827904877323287404567572, 10.51161540605892350177548527885
