| L(s) = 1 | − 3-s + 2·5-s + 9-s + 11-s − 2·15-s + 5·17-s + 7·19-s − 6·23-s − 25-s − 27-s + 9·29-s − 8·31-s − 33-s − 6·37-s − 2·41-s + 8·43-s + 2·45-s − 47-s − 5·51-s − 53-s + 2·55-s − 7·57-s + 9·59-s + 5·61-s + 3·67-s + 6·69-s + 15·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.301·11-s − 0.516·15-s + 1.21·17-s + 1.60·19-s − 1.25·23-s − 1/5·25-s − 0.192·27-s + 1.67·29-s − 1.43·31-s − 0.174·33-s − 0.986·37-s − 0.312·41-s + 1.21·43-s + 0.298·45-s − 0.145·47-s − 0.700·51-s − 0.137·53-s + 0.269·55-s − 0.927·57-s + 1.17·59-s + 0.640·61-s + 0.366·67-s + 0.722·69-s + 1.78·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99372 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99372 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.01802961070183, −13.71732749535495, −12.90997025995758, −12.54008863977370, −11.90408633102350, −11.77619073431914, −11.09502748911167, −10.33586824213547, −10.15775209918876, −9.646554093492100, −9.235437867735558, −8.583345189603044, −7.806858567582514, −7.603883983315213, −6.729472226508703, −6.472143100175402, −5.675033236489165, −5.379057278679409, −5.095735059748508, −3.983054216223103, −3.754077893589045, −2.874355872242822, −2.272019241669676, −1.406704442179513, −1.088115330071247, 0,
1.088115330071247, 1.406704442179513, 2.272019241669676, 2.874355872242822, 3.754077893589045, 3.983054216223103, 5.095735059748508, 5.379057278679409, 5.675033236489165, 6.472143100175402, 6.729472226508703, 7.603883983315213, 7.806858567582514, 8.583345189603044, 9.235437867735558, 9.646554093492100, 10.15775209918876, 10.33586824213547, 11.09502748911167, 11.77619073431914, 11.90408633102350, 12.54008863977370, 12.90997025995758, 13.71732749535495, 14.01802961070183
