| L(s) = 1 | − 3·5-s − 3·7-s − 11-s − 13-s + 5·17-s + 4·19-s − 8·23-s + 4·25-s + 9·35-s − 5·37-s + 11·43-s + 9·47-s + 2·49-s + 6·53-s + 3·55-s − 6·59-s + 3·65-s + 8·67-s + 15·71-s − 4·73-s + 3·77-s − 14·79-s + 8·83-s − 15·85-s + 12·89-s + 3·91-s − 12·95-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 1.13·7-s − 0.301·11-s − 0.277·13-s + 1.21·17-s + 0.917·19-s − 1.66·23-s + 4/5·25-s + 1.52·35-s − 0.821·37-s + 1.67·43-s + 1.31·47-s + 2/7·49-s + 0.824·53-s + 0.404·55-s − 0.781·59-s + 0.372·65-s + 0.977·67-s + 1.78·71-s − 0.468·73-s + 0.341·77-s − 1.57·79-s + 0.878·83-s − 1.62·85-s + 1.27·89-s + 0.314·91-s − 1.23·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−16.66724515339159, −16.18254511694861, −15.65874342185443, −15.57253907009543, −14.53286629152006, −14.09607510530623, −13.46119272661013, −12.62101657753818, −12.10212511618344, −12.01735814142823, −11.08209603808987, −10.41727599478672, −9.831879904748885, −9.329568260596725, −8.462309570027642, −7.762005335485104, −7.485058167984562, −6.742459901716455, −5.871798447402245, −5.364117182390377, −4.302036845492907, −3.718327189993172, −3.212972363399769, −2.336497844186559, −0.9126323005554363, 0,
0.9126323005554363, 2.336497844186559, 3.212972363399769, 3.718327189993172, 4.302036845492907, 5.364117182390377, 5.871798447402245, 6.742459901716455, 7.485058167984562, 7.762005335485104, 8.462309570027642, 9.329568260596725, 9.831879904748885, 10.41727599478672, 11.08209603808987, 12.01735814142823, 12.10212511618344, 12.62101657753818, 13.46119272661013, 14.09607510530623, 14.53286629152006, 15.57253907009543, 15.65874342185443, 16.18254511694861, 16.66724515339159
