| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 12-s + 13-s + 14-s + 16-s − 18-s − 2·19-s + 21-s − 3·23-s + 24-s − 26-s − 27-s − 28-s + 3·29-s + 8·31-s − 32-s + 36-s − 2·37-s + 2·38-s − 39-s + 9·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.458·19-s + 0.218·21-s − 0.625·23-s + 0.204·24-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + 0.557·29-s + 1.43·31-s − 0.176·32-s + 1/6·36-s − 0.328·37-s + 0.324·38-s − 0.160·39-s + 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 18 T + p T^{2} \) | 1.83.as |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.21042867821795, −12.42575498088338, −12.13765007417907, −11.81261070957513, −11.17869869334118, −10.66805350876384, −10.48007916000441, −9.853801361738037, −9.506899770706950, −8.929876044848997, −8.518292673986239, −7.834455007696603, −7.670408867431168, −6.833326681756979, −6.532971442739193, −6.064615184439388, −5.648601237577201, −4.969338790079057, −4.302133907152536, −3.996492183125084, −3.109827945233598, −2.676757619250754, −2.010090598665395, −1.287885109758835, −0.7227442770004411, 0,
0.7227442770004411, 1.287885109758835, 2.010090598665395, 2.676757619250754, 3.109827945233598, 3.996492183125084, 4.302133907152536, 4.969338790079057, 5.648601237577201, 6.064615184439388, 6.532971442739193, 6.833326681756979, 7.670408867431168, 7.834455007696603, 8.518292673986239, 8.929876044848997, 9.506899770706950, 9.853801361738037, 10.48007916000441, 10.66805350876384, 11.17869869334118, 11.81261070957513, 12.13765007417907, 12.42575498088338, 13.21042867821795
