| L(s) = 1 | − 3-s − 2·5-s − 7-s + 9-s + 2·15-s − 6·17-s − 4·19-s + 21-s − 25-s − 27-s − 2·29-s + 2·35-s + 2·37-s − 10·41-s − 8·43-s − 2·45-s + 49-s + 6·51-s − 2·53-s + 4·57-s + 12·59-s − 6·61-s − 63-s − 8·67-s − 8·71-s + 14·73-s + 75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.516·15-s − 1.45·17-s − 0.917·19-s + 0.218·21-s − 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.338·35-s + 0.328·37-s − 1.56·41-s − 1.21·43-s − 0.298·45-s + 1/7·49-s + 0.840·51-s − 0.274·53-s + 0.529·57-s + 1.56·59-s − 0.768·61-s − 0.125·63-s − 0.977·67-s − 0.949·71-s + 1.63·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 113568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 113568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.67120755033193, −13.21538300362400, −13.00907273438053, −12.30711198153127, −11.85022329433562, −11.49114623138608, −11.03788282660364, −10.53110347519290, −10.06342813239262, −9.504210602527497, −8.805544242299366, −8.509342184871910, −7.921934263238851, −7.302646708733536, −6.808864355812899, −6.431849140158074, −5.888077323761557, −5.161132625589590, −4.619054471369363, −4.171128621644977, −3.630988107595079, −3.022464107953457, −2.156617776484075, −1.662311263259684, −0.5343682202016477, 0,
0.5343682202016477, 1.662311263259684, 2.156617776484075, 3.022464107953457, 3.630988107595079, 4.171128621644977, 4.619054471369363, 5.161132625589590, 5.888077323761557, 6.431849140158074, 6.808864355812899, 7.302646708733536, 7.921934263238851, 8.509342184871910, 8.805544242299366, 9.504210602527497, 10.06342813239262, 10.53110347519290, 11.03788282660364, 11.49114623138608, 11.85022329433562, 12.30711198153127, 13.00907273438053, 13.21538300362400, 13.67120755033193
