| L(s) = 1 | − 2·5-s − 4·7-s − 11-s + 2·13-s + 2·17-s + 8·23-s − 25-s − 6·29-s + 8·31-s + 8·35-s − 6·37-s + 2·41-s + 8·47-s + 9·49-s + 6·53-s + 2·55-s + 4·59-s − 6·61-s − 4·65-s − 4·67-s − 14·73-s + 4·77-s + 4·79-s − 12·83-s − 4·85-s + 6·89-s − 8·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s − 0.301·11-s + 0.554·13-s + 0.485·17-s + 1.66·23-s − 1/5·25-s − 1.11·29-s + 1.43·31-s + 1.35·35-s − 0.986·37-s + 0.312·41-s + 1.16·47-s + 9/7·49-s + 0.824·53-s + 0.269·55-s + 0.520·59-s − 0.768·61-s − 0.496·65-s − 0.488·67-s − 1.63·73-s + 0.455·77-s + 0.450·79-s − 1.31·83-s − 0.433·85-s + 0.635·89-s − 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−7.42275429983940735099135130430, −7.18896686257792985441250548424, −6.25377810228368305041472746290, −5.68721248179579351211919147224, −4.72191662284056988692815425652, −3.81064897716730687178513490998, −3.31425171933473999322603981332, −2.57961275990012009877837343882, −1.06341230628835129193123862023, 0,
1.06341230628835129193123862023, 2.57961275990012009877837343882, 3.31425171933473999322603981332, 3.81064897716730687178513490998, 4.72191662284056988692815425652, 5.68721248179579351211919147224, 6.25377810228368305041472746290, 7.18896686257792985441250548424, 7.42275429983940735099135130430
