| L(s) = 1 | − 5-s − 7-s + 2·11-s − 4·13-s + 3·17-s + 2·19-s − 6·23-s − 4·25-s + 6·29-s − 4·31-s + 35-s − 11·37-s + 5·41-s − 7·43-s − 5·47-s + 49-s − 6·53-s − 2·55-s − 9·59-s + 6·61-s + 4·65-s − 4·67-s − 8·71-s − 2·73-s − 2·77-s − 11·79-s + 5·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 0.603·11-s − 1.10·13-s + 0.727·17-s + 0.458·19-s − 1.25·23-s − 4/5·25-s + 1.11·29-s − 0.718·31-s + 0.169·35-s − 1.80·37-s + 0.780·41-s − 1.06·43-s − 0.729·47-s + 1/7·49-s − 0.824·53-s − 0.269·55-s − 1.17·59-s + 0.768·61-s + 0.496·65-s − 0.488·67-s − 0.949·71-s − 0.234·73-s − 0.227·77-s − 1.23·79-s + 0.548·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 5 T + p T^{2} \) | 1.47.f |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 - 5 T + p T^{2} \) | 1.83.af |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−9.166364461823336339471398707217, −8.168700109725466244505479882013, −7.50859586020194953777192673110, −6.71248943603432804219900215576, −5.79626654899519538670964284688, −4.85551974950610032561351401695, −3.87880484327128264795072222540, −3.03742610634509483360169118311, −1.70503653258578344840698792805, 0,
1.70503653258578344840698792805, 3.03742610634509483360169118311, 3.87880484327128264795072222540, 4.85551974950610032561351401695, 5.79626654899519538670964284688, 6.71248943603432804219900215576, 7.50859586020194953777192673110, 8.168700109725466244505479882013, 9.166364461823336339471398707217
