| L(s) = 1 | + 2·5-s − 7-s + 5·11-s + 4·13-s − 2·17-s − 25-s + 29-s + 3·31-s − 2·35-s − 4·37-s + 2·43-s − 8·47-s − 6·49-s + 9·53-s + 10·55-s − 7·59-s − 4·61-s + 8·65-s + 8·67-s + 10·71-s − 3·73-s − 5·77-s − 3·79-s + 3·83-s − 4·85-s + 10·89-s − 4·91-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s + 1.50·11-s + 1.10·13-s − 0.485·17-s − 1/5·25-s + 0.185·29-s + 0.538·31-s − 0.338·35-s − 0.657·37-s + 0.304·43-s − 1.16·47-s − 6/7·49-s + 1.23·53-s + 1.34·55-s − 0.911·59-s − 0.512·61-s + 0.992·65-s + 0.977·67-s + 1.18·71-s − 0.351·73-s − 0.569·77-s − 0.337·79-s + 0.329·83-s − 0.433·85-s + 1.05·89-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.301522090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.301522090\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−14.79862259809504, −14.15060865679993, −13.82910797904418, −13.39227792066410, −12.85111536816451, −12.22727347086895, −11.69170955205326, −11.17414813217256, −10.64578252216155, −9.945993972444362, −9.544291175441442, −9.050152215062066, −8.548530208632045, −7.985233115815132, −7.023605764884764, −6.571107379462844, −6.196424512188321, −5.680772728376050, −4.882631881230982, −4.166018000169347, −3.612901134692555, −2.984329791028165, −1.996261799651685, −1.530053454989722, −0.6894379803283705,
0.6894379803283705, 1.530053454989722, 1.996261799651685, 2.984329791028165, 3.612901134692555, 4.166018000169347, 4.882631881230982, 5.680772728376050, 6.196424512188321, 6.571107379462844, 7.023605764884764, 7.985233115815132, 8.548530208632045, 9.050152215062066, 9.544291175441442, 9.945993972444362, 10.64578252216155, 11.17414813217256, 11.69170955205326, 12.22727347086895, 12.85111536816451, 13.39227792066410, 13.82910797904418, 14.15060865679993, 14.79862259809504
