| L(s) = 1 | + 2-s + 4-s − 4·5-s + 7-s + 8-s − 4·10-s + 11-s + 14-s + 16-s + 17-s + 4·19-s − 4·20-s + 22-s − 6·23-s + 11·25-s + 28-s − 6·29-s − 8·31-s + 32-s + 34-s − 4·35-s + 6·37-s + 4·38-s − 4·40-s + 2·41-s + 8·43-s + 44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.78·5-s + 0.377·7-s + 0.353·8-s − 1.26·10-s + 0.301·11-s + 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.917·19-s − 0.894·20-s + 0.213·22-s − 1.25·23-s + 11/5·25-s + 0.188·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s + 0.171·34-s − 0.676·35-s + 0.986·37-s + 0.648·38-s − 0.632·40-s + 0.312·41-s + 1.21·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23562 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23562 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−15.70765251537015, −15.05237654867416, −14.71572102754651, −14.25397107427856, −13.61058067112331, −12.88840840431328, −12.37535163573695, −11.91493008771049, −11.54432518251694, −10.95296607529278, −10.62406703246985, −9.557658461934510, −9.116391486734127, −8.232425686449160, −7.707347074424565, −7.482496628328613, −6.826038091371814, −5.892249391492713, −5.450132033844502, −4.535241389698002, −4.141149916303307, −3.598664965737025, −3.013568085690076, −2.019835805138384, −1.052409644678561, 0,
1.052409644678561, 2.019835805138384, 3.013568085690076, 3.598664965737025, 4.141149916303307, 4.535241389698002, 5.450132033844502, 5.892249391492713, 6.826038091371814, 7.482496628328613, 7.707347074424565, 8.232425686449160, 9.116391486734127, 9.557658461934510, 10.62406703246985, 10.95296607529278, 11.54432518251694, 11.91493008771049, 12.37535163573695, 12.88840840431328, 13.61058067112331, 14.25397107427856, 14.71572102754651, 15.05237654867416, 15.70765251537015
