| L(s) = 1 | − 2·5-s + 4·11-s − 13-s − 2·19-s − 4·23-s − 25-s − 2·31-s + 10·37-s + 2·41-s − 8·43-s + 12·53-s − 8·55-s − 12·59-s − 6·61-s + 2·65-s + 6·67-s + 8·71-s + 2·73-s + 12·79-s + 4·83-s + 14·89-s + 4·95-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.20·11-s − 0.277·13-s − 0.458·19-s − 0.834·23-s − 1/5·25-s − 0.359·31-s + 1.64·37-s + 0.312·41-s − 1.21·43-s + 1.64·53-s − 1.07·55-s − 1.56·59-s − 0.768·61-s + 0.248·65-s + 0.733·67-s + 0.949·71-s + 0.234·73-s + 1.35·79-s + 0.439·83-s + 1.48·89-s + 0.410·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 366912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.957906405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.957906405\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 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
−12.39609922530589, −11.93855384828395, −11.75993450087795, −11.17210137747756, −10.86814673428170, −10.19806066757568, −9.802346069663268, −9.216506339863087, −8.994451341834739, −8.289329547269619, −7.889827906784342, −7.600421439299667, −6.964815274587454, −6.447722775664765, −6.162111995008894, −5.535513741370787, −4.861580333962573, −4.390718252484512, −3.949878521155752, −3.587648394468037, −3.004255439942369, −2.184615228984330, −1.829920527916957, −0.9461865950319623, −0.4271591039614721,
0.4271591039614721, 0.9461865950319623, 1.829920527916957, 2.184615228984330, 3.004255439942369, 3.587648394468037, 3.949878521155752, 4.390718252484512, 4.861580333962573, 5.535513741370787, 6.162111995008894, 6.447722775664765, 6.964815274587454, 7.600421439299667, 7.889827906784342, 8.289329547269619, 8.994451341834739, 9.216506339863087, 9.802346069663268, 10.19806066757568, 10.86814673428170, 11.17210137747756, 11.75993450087795, 11.93855384828395, 12.39609922530589
