| L(s) = 1 | + 5-s − 3·9-s − 2·13-s − 6·17-s − 8·19-s + 25-s + 29-s + 4·31-s − 2·37-s − 6·41-s + 8·43-s − 3·45-s + 4·47-s − 7·49-s − 10·53-s + 4·59-s − 2·61-s − 2·65-s − 12·67-s − 8·71-s + 2·73-s + 4·79-s + 9·81-s + 12·83-s − 6·85-s − 6·89-s − 8·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 9-s − 0.554·13-s − 1.45·17-s − 1.83·19-s + 1/5·25-s + 0.185·29-s + 0.718·31-s − 0.328·37-s − 0.937·41-s + 1.21·43-s − 0.447·45-s + 0.583·47-s − 49-s − 1.37·53-s + 0.520·59-s − 0.256·61-s − 0.248·65-s − 1.46·67-s − 0.949·71-s + 0.234·73-s + 0.450·79-s + 81-s + 1.31·83-s − 0.650·85-s − 0.635·89-s − 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−9.185313026570586947706109540340, −8.714749446238551208220759932661, −7.85667511572974906227799583809, −6.62381303323909528238086721537, −6.18479486155746975507439983453, −5.04636316497160143255437022531, −4.23701838302685586091025961435, −2.83950161986258862285426873462, −2.01698592186055502906823834564, 0,
2.01698592186055502906823834564, 2.83950161986258862285426873462, 4.23701838302685586091025961435, 5.04636316497160143255437022531, 6.18479486155746975507439983453, 6.62381303323909528238086721537, 7.85667511572974906227799583809, 8.714749446238551208220759932661, 9.185313026570586947706109540340
