| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s − 2·11-s − 13-s + 16-s + 2·17-s + 4·19-s − 20-s − 2·22-s + 6·23-s + 25-s − 26-s − 7·29-s − 7·31-s + 32-s + 2·34-s − 5·37-s + 4·38-s − 40-s − 2·41-s + 5·43-s − 2·44-s + 6·46-s + 5·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 0.603·11-s − 0.277·13-s + 1/4·16-s + 0.485·17-s + 0.917·19-s − 0.223·20-s − 0.426·22-s + 1.25·23-s + 1/5·25-s − 0.196·26-s − 1.29·29-s − 1.25·31-s + 0.176·32-s + 0.342·34-s − 0.821·37-s + 0.648·38-s − 0.158·40-s − 0.312·41-s + 0.762·43-s − 0.301·44-s + 0.884·46-s + 0.729·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 263 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 5 T + p T^{2} \) | 1.47.af |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 7 T + p T^{2} \) | 1.83.h |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.58456830697209, −15.20731417714921, −14.50266372836294, −14.28467828922784, −13.42183035777986, −13.04382514255448, −12.54698328715400, −11.97612657251232, −11.42953312009398, −10.88583941173766, −10.46570282775457, −9.627164789772266, −9.156792668759966, −8.436483207119702, −7.579235385678201, −7.403435114733558, −6.794389842322710, −5.837297141838617, −5.369327188390318, −4.914326196316213, −4.090915002780269, −3.399456492520069, −2.959616234697542, −2.045785983856650, −1.171615489464078, 0,
1.171615489464078, 2.045785983856650, 2.959616234697542, 3.399456492520069, 4.090915002780269, 4.914326196316213, 5.369327188390318, 5.837297141838617, 6.794389842322710, 7.403435114733558, 7.579235385678201, 8.436483207119702, 9.156792668759966, 9.627164789772266, 10.46570282775457, 10.88583941173766, 11.42953312009398, 11.97612657251232, 12.54698328715400, 13.04382514255448, 13.42183035777986, 14.28467828922784, 14.50266372836294, 15.20731417714921, 15.58456830697209
