| L(s) = 1 | − 2·5-s + 3·11-s + 13-s + 5·17-s + 19-s − 25-s + 29-s + 4·31-s − 2·37-s + 10·41-s + 10·43-s + 47-s + 3·53-s − 6·55-s − 3·59-s + 5·61-s − 2·65-s − 5·67-s − 71-s − 12·73-s + 6·79-s − 16·83-s − 10·85-s − 14·89-s − 2·95-s − 4·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.904·11-s + 0.277·13-s + 1.21·17-s + 0.229·19-s − 1/5·25-s + 0.185·29-s + 0.718·31-s − 0.328·37-s + 1.56·41-s + 1.52·43-s + 0.145·47-s + 0.412·53-s − 0.809·55-s − 0.390·59-s + 0.640·61-s − 0.248·65-s − 0.610·67-s − 0.118·71-s − 1.40·73-s + 0.675·79-s − 1.75·83-s − 1.08·85-s − 1.48·89-s − 0.205·95-s − 0.406·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−14.04503150897493, −13.80249586428775, −13.01561167392718, −12.46117946837039, −12.11090987989892, −11.72008621040497, −11.16014711208180, −10.80515247643059, −9.925424670540321, −9.816476179101996, −8.900375314209099, −8.736850048350425, −7.894687639369599, −7.619980671900048, −7.136450732452461, −6.436239559941591, −5.888988409350960, −5.453003842948430, −4.615424184942911, −4.031688415101120, −3.803560820330173, −2.989383710324169, −2.487280404297525, −1.345394719239304, −1.021309364424408, 0,
1.021309364424408, 1.345394719239304, 2.487280404297525, 2.989383710324169, 3.803560820330173, 4.031688415101120, 4.615424184942911, 5.453003842948430, 5.888988409350960, 6.436239559941591, 7.136450732452461, 7.619980671900048, 7.894687639369599, 8.736850048350425, 8.900375314209099, 9.816476179101996, 9.925424670540321, 10.80515247643059, 11.16014711208180, 11.72008621040497, 12.11090987989892, 12.46117946837039, 13.01561167392718, 13.80249586428775, 14.04503150897493
