| L(s) = 1 | − 3-s + 2·5-s − 4·7-s + 9-s − 11-s − 2·15-s + 2·17-s + 4·19-s + 4·21-s − 25-s − 27-s + 6·29-s − 4·31-s + 33-s − 8·35-s − 2·37-s + 6·41-s + 4·43-s + 2·45-s + 8·47-s + 9·49-s − 2·51-s + 6·53-s − 2·55-s − 4·57-s + 12·59-s + 10·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 0.516·15-s + 0.485·17-s + 0.917·19-s + 0.872·21-s − 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.718·31-s + 0.174·33-s − 1.35·35-s − 0.328·37-s + 0.937·41-s + 0.609·43-s + 0.298·45-s + 1.16·47-s + 9/7·49-s − 0.280·51-s + 0.824·53-s − 0.269·55-s − 0.529·57-s + 1.56·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 356928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 356928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−12.72770037205352, −12.35854326181597, −11.95139997918684, −11.46829926779363, −10.76718550805488, −10.47260179370457, −9.953336740713929, −9.690373595211355, −9.334318359688358, −8.784453576083061, −8.228544190132745, −7.453635277963733, −7.226758466698748, −6.610645181745117, −6.249290670318226, −5.738301493607519, −5.404589130818978, −5.031840809697659, −4.047984978815671, −3.818789671118311, −3.120464948632175, −2.531211445034740, −2.212729456280405, −1.181137581038600, −0.8121800273849353, 0,
0.8121800273849353, 1.181137581038600, 2.212729456280405, 2.531211445034740, 3.120464948632175, 3.818789671118311, 4.047984978815671, 5.031840809697659, 5.404589130818978, 5.738301493607519, 6.249290670318226, 6.610645181745117, 7.226758466698748, 7.453635277963733, 8.228544190132745, 8.784453576083061, 9.334318359688358, 9.690373595211355, 9.953336740713929, 10.47260179370457, 10.76718550805488, 11.46829926779363, 11.95139997918684, 12.35854326181597, 12.72770037205352
