| L(s) = 1 | − 3-s − 2·5-s + 9-s − 13-s + 2·15-s + 4·17-s + 8·19-s − 6·23-s − 25-s − 27-s − 2·29-s − 8·31-s + 2·37-s + 39-s + 2·41-s − 4·43-s − 2·45-s + 2·47-s − 4·51-s + 6·53-s − 8·57-s + 6·59-s − 2·61-s + 2·65-s − 4·67-s + 6·69-s + 4·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.277·13-s + 0.516·15-s + 0.970·17-s + 1.83·19-s − 1.25·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.328·37-s + 0.160·39-s + 0.312·41-s − 0.609·43-s − 0.298·45-s + 0.291·47-s − 0.560·51-s + 0.824·53-s − 1.05·57-s + 0.781·59-s − 0.256·61-s + 0.248·65-s − 0.488·67-s + 0.722·69-s + 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.85054244765326, −13.29213788074924, −12.51312455960744, −12.39864046397760, −11.74283951697974, −11.47052422822625, −11.13829561851228, −10.23908775533247, −10.04191553667618, −9.485870410085499, −8.965680730105775, −8.179302838978735, −7.804638516443247, −7.323507625718811, −7.079401731314768, −6.176762013218411, −5.621093488775213, −5.370557383440341, −4.681132553896394, −3.955968876664120, −3.642346407568614, −3.057283354496233, −2.193498837557597, −1.456458055614032, −0.7368502283126629, 0,
0.7368502283126629, 1.456458055614032, 2.193498837557597, 3.057283354496233, 3.642346407568614, 3.955968876664120, 4.681132553896394, 5.370557383440341, 5.621093488775213, 6.176762013218411, 7.079401731314768, 7.323507625718811, 7.804638516443247, 8.179302838978735, 8.965680730105775, 9.485870410085499, 10.04191553667618, 10.23908775533247, 11.13829561851228, 11.47052422822625, 11.74283951697974, 12.39864046397760, 12.51312455960744, 13.29213788074924, 13.85054244765326
