| L(s) = 1 | − 2-s + 3-s − 4-s − 6-s + 3·8-s + 9-s − 12-s − 6·13-s − 16-s + 2·17-s − 18-s + 8·19-s − 8·23-s + 3·24-s + 6·26-s + 27-s − 2·29-s − 4·31-s − 5·32-s − 2·34-s − 36-s + 2·37-s − 8·38-s − 6·39-s + 6·41-s − 4·43-s + 8·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.288·12-s − 1.66·13-s − 1/4·16-s + 0.485·17-s − 0.235·18-s + 1.83·19-s − 1.66·23-s + 0.612·24-s + 1.17·26-s + 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.883·32-s − 0.342·34-s − 1/6·36-s + 0.328·37-s − 1.29·38-s − 0.960·39-s + 0.937·41-s − 0.609·43-s + 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−8.006648689764812283420657294904, −7.66078399107701284311775872096, −7.16566744133333875383333517560, −5.82680289170118734614021714302, −5.07896009712412898647893372722, −4.30076158847618591361657861041, −3.40922618253584977154524124894, −2.38993500725665786110797187633, −1.34577586096679437279208168488, 0,
1.34577586096679437279208168488, 2.38993500725665786110797187633, 3.40922618253584977154524124894, 4.30076158847618591361657861041, 5.07896009712412898647893372722, 5.82680289170118734614021714302, 7.16566744133333875383333517560, 7.66078399107701284311775872096, 8.006648689764812283420657294904
