| L(s) = 1 | − 3-s − 2·4-s + 3·5-s − 2·9-s + 11-s + 2·12-s − 3·15-s + 4·16-s + 6·17-s + 2·19-s − 6·20-s + 3·23-s + 4·25-s + 5·27-s − 6·29-s + 5·31-s − 33-s + 4·36-s − 11·37-s + 6·41-s + 8·43-s − 2·44-s − 6·45-s − 4·48-s − 6·51-s − 6·53-s + 3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 1.34·5-s − 2/3·9-s + 0.301·11-s + 0.577·12-s − 0.774·15-s + 16-s + 1.45·17-s + 0.458·19-s − 1.34·20-s + 0.625·23-s + 4/5·25-s + 0.962·27-s − 1.11·29-s + 0.898·31-s − 0.174·33-s + 2/3·36-s − 1.80·37-s + 0.937·41-s + 1.21·43-s − 0.301·44-s − 0.894·45-s − 0.577·48-s − 0.840·51-s − 0.824·53-s + 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91091 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91091 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−14.17048332260262, −13.62642322299007, −13.21273321907165, −12.62444112029577, −12.19991164532904, −11.76720433175091, −11.03354110142044, −10.54658888000605, −10.09669473342681, −9.606398828956924, −9.149575265054418, −8.840595271756779, −8.108480321257307, −7.582702370777847, −6.935467226360111, −6.135313230290786, −5.822748354821596, −5.422476637203490, −5.001712552955741, −4.359360724144139, −3.485552438567617, −3.101824184360303, −2.297398938663782, −1.383269768584841, −0.9658437076252255, 0,
0.9658437076252255, 1.383269768584841, 2.297398938663782, 3.101824184360303, 3.485552438567617, 4.359360724144139, 5.001712552955741, 5.422476637203490, 5.822748354821596, 6.135313230290786, 6.935467226360111, 7.582702370777847, 8.108480321257307, 8.840595271756779, 9.149575265054418, 9.606398828956924, 10.09669473342681, 10.54658888000605, 11.03354110142044, 11.76720433175091, 12.19991164532904, 12.62444112029577, 13.21273321907165, 13.62642322299007, 14.17048332260262
