| L(s) = 1 | + 2·5-s + 11-s + 2·13-s + 3·17-s + 23-s − 25-s − 7·29-s + 2·31-s − 6·37-s + 6·41-s − 6·43-s − 3·47-s − 12·53-s + 2·55-s − 4·59-s − 10·61-s + 4·65-s + 8·67-s − 13·71-s + 11·73-s + 13·79-s − 4·83-s + 6·85-s + 6·89-s + 10·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.301·11-s + 0.554·13-s + 0.727·17-s + 0.208·23-s − 1/5·25-s − 1.29·29-s + 0.359·31-s − 0.986·37-s + 0.937·41-s − 0.914·43-s − 0.437·47-s − 1.64·53-s + 0.269·55-s − 0.520·59-s − 1.28·61-s + 0.496·65-s + 0.977·67-s − 1.54·71-s + 1.28·73-s + 1.46·79-s − 0.439·83-s + 0.650·85-s + 0.635·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 13 T + p T^{2} \) | 1.71.n |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−13.51597740081477, −13.06449783431843, −12.60790395039690, −12.15529288240006, −11.52044946999147, −11.16469774357528, −10.60163338681421, −10.14173106745864, −9.642714724968609, −9.245710839437847, −8.848966805334147, −8.163779580192320, −7.693127086033105, −7.245623966459330, −6.416912021533026, −6.230775818935691, −5.691398405257173, −5.100796289944302, −4.690247573327426, −3.819883980003282, −3.444927590265694, −2.853666151890193, −1.976160734526206, −1.668364013657448, −0.9605245265306303, 0,
0.9605245265306303, 1.668364013657448, 1.976160734526206, 2.853666151890193, 3.444927590265694, 3.819883980003282, 4.690247573327426, 5.100796289944302, 5.691398405257173, 6.230775818935691, 6.416912021533026, 7.245623966459330, 7.693127086033105, 8.163779580192320, 8.848966805334147, 9.245710839437847, 9.642714724968609, 10.14173106745864, 10.60163338681421, 11.16469774357528, 11.52044946999147, 12.15529288240006, 12.60790395039690, 13.06449783431843, 13.51597740081477
