| L(s) = 1 | − 5-s + 3·7-s + 3·11-s + 2·17-s − 6·19-s − 3·23-s − 4·25-s + 4·29-s − 6·31-s − 3·35-s − 7·37-s − 5·41-s − 6·43-s + 9·47-s + 2·49-s − 11·53-s − 3·55-s + 6·59-s − 8·61-s − 12·71-s − 15·73-s + 9·77-s + 12·79-s + 3·83-s − 2·85-s − 2·89-s + 6·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.13·7-s + 0.904·11-s + 0.485·17-s − 1.37·19-s − 0.625·23-s − 4/5·25-s + 0.742·29-s − 1.07·31-s − 0.507·35-s − 1.15·37-s − 0.780·41-s − 0.914·43-s + 1.31·47-s + 2/7·49-s − 1.51·53-s − 0.404·55-s + 0.781·59-s − 1.02·61-s − 1.42·71-s − 1.75·73-s + 1.02·77-s + 1.35·79-s + 0.329·83-s − 0.216·85-s − 0.211·89-s + 0.615·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.167576765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.167576765\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 15 T + p T^{2} \) | 1.73.p |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−12.89009399245431, −12.25336542761409, −11.96331238683143, −11.69848211462284, −11.08724569508931, −10.58898439979437, −10.35110604460507, −9.618580263429190, −9.109558048903753, −8.613963362027792, −8.241555005894499, −7.811786979358202, −7.311264955938556, −6.721744039107639, −6.299871818220463, −5.669226987774489, −5.195840213693641, −4.532946121133794, −4.193350730165938, −3.692126691803839, −3.115945976217873, −2.273402548566854, −1.629604622453049, −1.429927381103674, −0.2875167341102733,
0.2875167341102733, 1.429927381103674, 1.629604622453049, 2.273402548566854, 3.115945976217873, 3.692126691803839, 4.193350730165938, 4.532946121133794, 5.195840213693641, 5.669226987774489, 6.299871818220463, 6.721744039107639, 7.311264955938556, 7.811786979358202, 8.241555005894499, 8.613963362027792, 9.109558048903753, 9.618580263429190, 10.35110604460507, 10.58898439979437, 11.08724569508931, 11.69848211462284, 11.96331238683143, 12.25336542761409, 12.89009399245431
