| L(s) = 1 | + 2·5-s + 7-s − 4·11-s + 13-s − 6·17-s + 4·19-s − 25-s − 6·29-s − 8·31-s + 2·35-s − 10·37-s + 6·41-s − 4·43-s − 4·47-s + 49-s + 10·53-s − 8·55-s + 4·59-s + 6·61-s + 2·65-s + 8·67-s − 10·73-s − 4·77-s − 8·79-s + 4·83-s − 12·85-s + 6·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s − 1.20·11-s + 0.277·13-s − 1.45·17-s + 0.917·19-s − 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.338·35-s − 1.64·37-s + 0.937·41-s − 0.609·43-s − 0.583·47-s + 1/7·49-s + 1.37·53-s − 1.07·55-s + 0.520·59-s + 0.768·61-s + 0.248·65-s + 0.977·67-s − 1.17·73-s − 0.455·77-s − 0.900·79-s + 0.439·83-s − 1.30·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.725222411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.725222411\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.34681327677523, −13.94272406457671, −13.32840091766881, −13.07796555939376, −12.69100731116018, −11.71796500147519, −11.41127021824876, −10.83477707858451, −10.30808553394766, −9.937751288310324, −9.060963821261051, −8.986292995391081, −8.189684505291860, −7.584858034225834, −7.097051619776797, −6.531383186335686, −5.650966477401667, −5.460048319076159, −4.940257374957164, −4.079203592580728, −3.498487462149274, −2.642973929531742, −2.047042208869259, −1.622795215873118, −0.4300529851582359,
0.4300529851582359, 1.622795215873118, 2.047042208869259, 2.642973929531742, 3.498487462149274, 4.079203592580728, 4.940257374957164, 5.460048319076159, 5.650966477401667, 6.531383186335686, 7.097051619776797, 7.584858034225834, 8.189684505291860, 8.986292995391081, 9.060963821261051, 9.937751288310324, 10.30808553394766, 10.83477707858451, 11.41127021824876, 11.71796500147519, 12.69100731116018, 13.07796555939376, 13.32840091766881, 13.94272406457671, 14.34681327677523
