| L(s) = 1 | − 2·3-s + 2·5-s + 9-s − 2·13-s − 4·15-s − 8·17-s − 4·23-s − 25-s + 4·27-s + 2·29-s + 6·31-s + 37-s + 4·39-s − 6·41-s − 4·43-s + 2·45-s + 8·47-s + 16·51-s − 6·53-s − 4·59-s − 10·61-s − 4·65-s − 16·67-s + 8·69-s − 8·71-s − 2·73-s + 2·75-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 1/3·9-s − 0.554·13-s − 1.03·15-s − 1.94·17-s − 0.834·23-s − 1/5·25-s + 0.769·27-s + 0.371·29-s + 1.07·31-s + 0.164·37-s + 0.640·39-s − 0.937·41-s − 0.609·43-s + 0.298·45-s + 1.16·47-s + 2.24·51-s − 0.824·53-s − 0.520·59-s − 1.28·61-s − 0.496·65-s − 1.95·67-s + 0.963·69-s − 0.949·71-s − 0.234·73-s + 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 16 T + p T^{2} \) | 1.67.q |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 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
−13.92018420141225, −13.59604110814226, −13.17145028609443, −12.53758975774803, −12.05099986160713, −11.69661763670489, −11.22363166266016, −10.61696150407352, −10.29953716377830, −9.859839290469920, −9.197695462921391, −8.802527020363780, −8.207683849740209, −7.553418599377573, −6.827708035108657, −6.526718863414860, −6.018548662555915, −5.649688963235687, −5.006223554075203, −4.442403133840210, −4.190663989327525, −2.956789909031492, −2.609362544305302, −1.807763063663938, −1.297649259350403, 0, 0,
1.297649259350403, 1.807763063663938, 2.609362544305302, 2.956789909031492, 4.190663989327525, 4.442403133840210, 5.006223554075203, 5.649688963235687, 6.018548662555915, 6.526718863414860, 6.827708035108657, 7.553418599377573, 8.207683849740209, 8.802527020363780, 9.197695462921391, 9.859839290469920, 10.29953716377830, 10.61696150407352, 11.22363166266016, 11.69661763670489, 12.05099986160713, 12.53758975774803, 13.17145028609443, 13.59604110814226, 13.92018420141225
