| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 12-s − 13-s − 15-s + 16-s + 6·17-s − 18-s − 19-s + 20-s + 24-s + 25-s + 26-s − 27-s + 6·29-s + 30-s + 4·31-s − 32-s − 6·34-s + 36-s + 2·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.277·13-s − 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 1.11·29-s + 0.182·30-s + 0.718·31-s − 0.176·32-s − 1.02·34-s + 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| 19 | \( 1 + T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.58714476056244, −12.16649661625518, −11.88660162904151, −11.29112320155354, −10.95234435092320, −10.23026502114903, −10.06163934987649, −9.797350766576368, −9.184605975547039, −8.581138280897626, −8.207536384041536, −7.787029884462849, −7.101840483619252, −6.881960433906404, −6.143418770069835, −5.967231449065825, −5.317548788672770, −4.851413207176355, −4.374372460864895, −3.583149399464494, −3.032677938444629, −2.600819887333781, −1.784300821621843, −1.332192110233821, −0.7664445347877116, 0,
0.7664445347877116, 1.332192110233821, 1.784300821621843, 2.600819887333781, 3.032677938444629, 3.583149399464494, 4.374372460864895, 4.851413207176355, 5.317548788672770, 5.967231449065825, 6.143418770069835, 6.881960433906404, 7.101840483619252, 7.787029884462849, 8.207536384041536, 8.581138280897626, 9.184605975547039, 9.797350766576368, 10.06163934987649, 10.23026502114903, 10.95234435092320, 11.29112320155354, 11.88660162904151, 12.16649661625518, 12.58714476056244
