| L(s) = 1 | − 3-s − 2·7-s − 2·9-s + 4·11-s − 13-s + 4·19-s + 2·21-s − 23-s + 5·27-s − 7·29-s + 7·31-s − 4·33-s + 4·37-s + 39-s + 3·41-s + 6·43-s − 13·47-s − 3·49-s − 10·53-s − 4·57-s + 8·59-s + 4·63-s + 8·67-s + 69-s − 13·71-s − 11·73-s − 8·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s − 2/3·9-s + 1.20·11-s − 0.277·13-s + 0.917·19-s + 0.436·21-s − 0.208·23-s + 0.962·27-s − 1.29·29-s + 1.25·31-s − 0.696·33-s + 0.657·37-s + 0.160·39-s + 0.468·41-s + 0.914·43-s − 1.89·47-s − 3/7·49-s − 1.37·53-s − 0.529·57-s + 1.04·59-s + 0.503·63-s + 0.977·67-s + 0.120·69-s − 1.54·71-s − 1.28·73-s − 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.185706996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.185706996\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 13 T + p T^{2} \) | 1.47.n |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 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.l |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−7.61940791046419599456545400698, −6.91521154487784881211638869862, −6.20961120043075875826480798209, −5.89221682100009039605211075706, −5.00245127904215320460802962187, −4.25840134978148469134736139450, −3.38208311758819130817883013306, −2.79426758675998621335804481523, −1.58729958647584751385589286707, −0.55005901347216561431807694969,
0.55005901347216561431807694969, 1.58729958647584751385589286707, 2.79426758675998621335804481523, 3.38208311758819130817883013306, 4.25840134978148469134736139450, 5.00245127904215320460802962187, 5.89221682100009039605211075706, 6.20961120043075875826480798209, 6.91521154487784881211638869862, 7.61940791046419599456545400698
