| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s − 7-s + 8-s + 9-s + 5·11-s − 2·12-s − 7·13-s − 14-s + 16-s + 18-s − 7·19-s + 2·21-s + 5·22-s + 23-s − 2·24-s − 7·26-s + 4·27-s − 28-s + 5·29-s − 10·31-s + 32-s − 10·33-s + 36-s − 2·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.50·11-s − 0.577·12-s − 1.94·13-s − 0.267·14-s + 1/4·16-s + 0.235·18-s − 1.60·19-s + 0.436·21-s + 1.06·22-s + 0.208·23-s − 0.408·24-s − 1.37·26-s + 0.769·27-s − 0.188·28-s + 0.928·29-s − 1.79·31-s + 0.176·32-s − 1.74·33-s + 1/6·36-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 17 T + p T^{2} \) | 1.83.r |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−9.597432256668881742495139167686, −8.627804380295061875284664152200, −7.29484271368926743487815004271, −6.61309405130226386181755674891, −6.07981170530492068497451174291, −4.98959050725611850944044633525, −4.44264405651524113529607292049, −3.21966337768408157347843564951, −1.85622918971842028675810874009, 0,
1.85622918971842028675810874009, 3.21966337768408157347843564951, 4.44264405651524113529607292049, 4.98959050725611850944044633525, 6.07981170530492068497451174291, 6.61309405130226386181755674891, 7.29484271368926743487815004271, 8.627804380295061875284664152200, 9.597432256668881742495139167686
