| L(s) = 1 | + 2-s + 3-s − 4-s − 5-s + 6-s − 3·8-s + 9-s − 10-s − 12-s + 4·13-s − 15-s − 16-s + 18-s + 4·19-s + 20-s − 8·23-s − 3·24-s + 25-s + 4·26-s + 27-s + 2·29-s − 30-s − 2·31-s + 5·32-s − 36-s − 6·37-s + 4·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 1.10·13-s − 0.258·15-s − 1/4·16-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 1.66·23-s − 0.612·24-s + 1/5·25-s + 0.784·26-s + 0.192·27-s + 0.371·29-s − 0.182·30-s − 0.359·31-s + 0.883·32-s − 1/6·36-s − 0.986·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 290445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 290445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.401777676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.401777676\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| 67 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−12.68651297808055, −12.34434214487520, −11.93857726289620, −11.53524278649439, −10.94118253500520, −10.30578315632434, −10.04485387042069, −9.368959160759836, −8.925966255836814, −8.641133614408281, −8.069232669107068, −7.687323207568282, −7.175361619486620, −6.389767744926413, −6.135473629712899, −5.503452685862690, −5.069109566048698, −4.424501992631076, −3.916728514937491, −3.726560353581363, −2.998823656590948, −2.741908519870123, −1.652500254761419, −1.327707918092562, −0.2697018334615790,
0.2697018334615790, 1.327707918092562, 1.652500254761419, 2.741908519870123, 2.998823656590948, 3.726560353581363, 3.916728514937491, 4.424501992631076, 5.069109566048698, 5.503452685862690, 6.135473629712899, 6.389767744926413, 7.175361619486620, 7.687323207568282, 8.069232669107068, 8.641133614408281, 8.925966255836814, 9.368959160759836, 10.04485387042069, 10.30578315632434, 10.94118253500520, 11.53524278649439, 11.93857726289620, 12.34434214487520, 12.68651297808055
