| L(s) = 1 | + 5-s − 2·7-s − 3·9-s − 2·11-s + 6·13-s + 2·17-s + 2·19-s − 6·23-s + 25-s + 29-s − 6·31-s − 2·35-s + 2·37-s + 10·41-s + 8·43-s − 3·45-s − 4·47-s − 3·49-s − 10·53-s − 2·55-s − 8·59-s − 10·61-s + 6·63-s + 6·65-s − 2·67-s + 4·71-s + 6·73-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 9-s − 0.603·11-s + 1.66·13-s + 0.485·17-s + 0.458·19-s − 1.25·23-s + 1/5·25-s + 0.185·29-s − 1.07·31-s − 0.338·35-s + 0.328·37-s + 1.56·41-s + 1.21·43-s − 0.447·45-s − 0.583·47-s − 3/7·49-s − 1.37·53-s − 0.269·55-s − 1.04·59-s − 1.28·61-s + 0.755·63-s + 0.744·65-s − 0.244·67-s + 0.474·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 29 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−7.54316214623180643227282195992, −6.36260452369456678804440951275, −6.01291732964671592776065378263, −5.61515533874963332999619153289, −4.60296901995440598762836703730, −3.60122572819278398000004182225, −3.14265639881187720092447194652, −2.26850982369866317528265345263, −1.20564476238376903112240910209, 0,
1.20564476238376903112240910209, 2.26850982369866317528265345263, 3.14265639881187720092447194652, 3.60122572819278398000004182225, 4.60296901995440598762836703730, 5.61515533874963332999619153289, 6.01291732964671592776065378263, 6.36260452369456678804440951275, 7.54316214623180643227282195992
