| L(s) = 1 | + 4·5-s + 7-s + 2·11-s − 4·13-s − 19-s − 6·23-s + 11·25-s + 10·29-s + 4·35-s − 6·37-s + 10·41-s + 8·43-s + 12·47-s + 49-s − 6·53-s + 8·55-s + 12·59-s + 2·61-s − 16·65-s − 2·67-s − 12·71-s − 6·73-s + 2·77-s − 2·79-s + 2·89-s − 4·91-s − 4·95-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.377·7-s + 0.603·11-s − 1.10·13-s − 0.229·19-s − 1.25·23-s + 11/5·25-s + 1.85·29-s + 0.676·35-s − 0.986·37-s + 1.56·41-s + 1.21·43-s + 1.75·47-s + 1/7·49-s − 0.824·53-s + 1.07·55-s + 1.56·59-s + 0.256·61-s − 1.98·65-s − 0.244·67-s − 1.42·71-s − 0.702·73-s + 0.227·77-s − 0.225·79-s + 0.211·89-s − 0.419·91-s − 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.353946220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.353946220\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 19 | \( 1 + T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−14.10082977468078, −13.72265387674709, −13.10221038246246, −12.45103163645413, −12.22101247946716, −11.66294625119060, −10.80626646641012, −10.49046152144780, −9.904909423282864, −9.690837779767759, −8.936176736022018, −8.715272301424462, −7.917589158778408, −7.264154370877330, −6.789451043497804, −6.161186853639744, −5.745637057307158, −5.320503721984352, −4.442715189801095, −4.292960657254866, −3.132237795429065, −2.445999083016592, −2.160901655696208, −1.394607233700173, −0.6929320604490985,
0.6929320604490985, 1.394607233700173, 2.160901655696208, 2.445999083016592, 3.132237795429065, 4.292960657254866, 4.442715189801095, 5.320503721984352, 5.745637057307158, 6.161186853639744, 6.789451043497804, 7.264154370877330, 7.917589158778408, 8.715272301424462, 8.936176736022018, 9.690837779767759, 9.904909423282864, 10.49046152144780, 10.80626646641012, 11.66294625119060, 12.22101247946716, 12.45103163645413, 13.10221038246246, 13.72265387674709, 14.10082977468078
