| L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s + 4·11-s − 14-s + 16-s − 6·19-s + 20-s − 4·22-s + 6·23-s + 25-s + 28-s − 2·29-s − 10·31-s − 32-s + 35-s + 10·37-s + 6·38-s − 40-s − 2·41-s − 2·43-s + 4·44-s − 6·46-s − 12·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s + 1.20·11-s − 0.267·14-s + 1/4·16-s − 1.37·19-s + 0.223·20-s − 0.852·22-s + 1.25·23-s + 1/5·25-s + 0.188·28-s − 0.371·29-s − 1.79·31-s − 0.176·32-s + 0.169·35-s + 1.64·37-s + 0.973·38-s − 0.158·40-s − 0.312·41-s − 0.304·43-s + 0.603·44-s − 0.884·46-s − 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.16566441967339, −13.22263629577716, −12.96418435004594, −12.58611846885962, −11.72304898655418, −11.46784856545630, −10.92060354175752, −10.62313410751085, −9.842685088904453, −9.478177595561561, −9.000423310718081, −8.603104369919935, −8.105210851596452, −7.366753406175608, −6.996411232221412, −6.391200854346974, −6.043736011191771, −5.330774153467139, −4.729035521087278, −4.101042856636293, −3.508594116614491, −2.806636338148306, −2.032764960667865, −1.614566369101083, −0.9479226731428305, 0,
0.9479226731428305, 1.614566369101083, 2.032764960667865, 2.806636338148306, 3.508594116614491, 4.101042856636293, 4.729035521087278, 5.330774153467139, 6.043736011191771, 6.391200854346974, 6.996411232221412, 7.366753406175608, 8.105210851596452, 8.603104369919935, 9.000423310718081, 9.478177595561561, 9.842685088904453, 10.62313410751085, 10.92060354175752, 11.46784856545630, 11.72304898655418, 12.58611846885962, 12.96418435004594, 13.22263629577716, 14.16566441967339
