| L(s) = 1 | + 3-s + 2·5-s + 2·7-s + 9-s + 4·11-s − 2·13-s + 2·15-s − 2·17-s − 4·19-s + 2·21-s + 2·23-s − 25-s + 27-s + 10·29-s − 4·31-s + 4·33-s + 4·35-s − 8·37-s − 2·39-s + 6·41-s + 2·45-s + 2·47-s − 3·49-s − 2·51-s + 12·53-s + 8·55-s − 4·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.516·15-s − 0.485·17-s − 0.917·19-s + 0.436·21-s + 0.417·23-s − 1/5·25-s + 0.192·27-s + 1.85·29-s − 0.718·31-s + 0.696·33-s + 0.676·35-s − 1.31·37-s − 0.320·39-s + 0.937·41-s + 0.298·45-s + 0.291·47-s − 3/7·49-s − 0.280·51-s + 1.64·53-s + 1.07·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 355008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 355008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 43 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−12.81323897788527, −12.25646028384406, −11.94630268676311, −11.45206178822422, −10.83171549048144, −10.45589283548553, −10.06849707746114, −9.481663142911501, −9.088039457081202, −8.666697685353148, −8.422998032218269, −7.694480232628946, −7.198877311188641, −6.777163552115886, −6.298918382764394, −5.837775223072563, −5.201975909992990, −4.730251588514322, −4.154653308365441, −3.926390764660287, −2.909398218693346, −2.674700512625653, −1.873247520161999, −1.628188186069292, −1.008005896304336, 0,
1.008005896304336, 1.628188186069292, 1.873247520161999, 2.674700512625653, 2.909398218693346, 3.926390764660287, 4.154653308365441, 4.730251588514322, 5.201975909992990, 5.837775223072563, 6.298918382764394, 6.777163552115886, 7.198877311188641, 7.694480232628946, 8.422998032218269, 8.666697685353148, 9.088039457081202, 9.481663142911501, 10.06849707746114, 10.45589283548553, 10.83171549048144, 11.45206178822422, 11.94630268676311, 12.25646028384406, 12.81323897788527
