| L(s) = 1 | − 2·4-s + 3·5-s − 7-s − 11-s + 4·16-s + 6·17-s − 2·19-s − 6·20-s − 3·23-s + 4·25-s + 2·28-s + 6·29-s − 5·31-s − 3·35-s − 11·37-s + 6·41-s + 8·43-s + 2·44-s + 49-s + 6·53-s − 3·55-s − 9·59-s − 10·61-s − 8·64-s − 5·67-s − 12·68-s + 9·71-s + ⋯ |
| L(s) = 1 | − 4-s + 1.34·5-s − 0.377·7-s − 0.301·11-s + 16-s + 1.45·17-s − 0.458·19-s − 1.34·20-s − 0.625·23-s + 4/5·25-s + 0.377·28-s + 1.11·29-s − 0.898·31-s − 0.507·35-s − 1.80·37-s + 0.937·41-s + 1.21·43-s + 0.301·44-s + 1/7·49-s + 0.824·53-s − 0.404·55-s − 1.17·59-s − 1.28·61-s − 64-s − 0.610·67-s − 1.45·68-s + 1.06·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.989315539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.989315539\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−13.67347258155880, −13.22701128222920, −12.57063052702149, −12.35202551914798, −11.94784255311729, −10.83666787444944, −10.54780297688468, −10.11169760300887, −9.699761647302977, −9.204168434509278, −8.869974447449130, −8.230711508992359, −7.687497999739681, −7.187813237318134, −6.335337098580224, −5.985792650739265, −5.476815486494677, −5.139401244308569, −4.410446990329121, −3.829347425445173, −3.189109352137698, −2.638404398754999, −1.837587134439326, −1.264214189252914, −0.4554052468304249,
0.4554052468304249, 1.264214189252914, 1.837587134439326, 2.638404398754999, 3.189109352137698, 3.829347425445173, 4.410446990329121, 5.139401244308569, 5.476815486494677, 5.985792650739265, 6.335337098580224, 7.187813237318134, 7.687497999739681, 8.230711508992359, 8.869974447449130, 9.204168434509278, 9.699761647302977, 10.11169760300887, 10.54780297688468, 10.83666787444944, 11.94784255311729, 12.35202551914798, 12.57063052702149, 13.22701128222920, 13.67347258155880
