| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 4·7-s + 8-s + 9-s − 11-s + 12-s − 4·14-s + 16-s − 2·17-s + 18-s − 4·19-s − 4·21-s − 22-s + 24-s + 27-s − 4·28-s + 2·29-s − 8·31-s + 32-s − 33-s − 2·34-s + 36-s − 6·37-s − 4·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.917·19-s − 0.872·21-s − 0.213·22-s + 0.204·24-s + 0.192·27-s − 0.755·28-s + 0.371·29-s − 1.43·31-s + 0.176·32-s − 0.174·33-s − 0.342·34-s + 1/6·36-s − 0.986·37-s − 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 278850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 278850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
| 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
−13.04383521920841, −12.65420979833123, −12.29671305945430, −11.76679452723744, −11.05560883852347, −10.73541847744766, −10.15313908233991, −9.908377885826377, −9.177229544924181, −8.876480915477667, −8.466361179978750, −7.697626654434459, −7.174143779364347, −6.982772284040608, −6.224664620334985, −6.033177115768474, −5.393974072220876, −4.789951927528502, −4.103767584376391, −3.862187228452697, −3.203108444329289, −2.826964900845198, −2.225595315613885, −1.760727170045141, −0.7321397911764336, 0,
0.7321397911764336, 1.760727170045141, 2.225595315613885, 2.826964900845198, 3.203108444329289, 3.862187228452697, 4.103767584376391, 4.789951927528502, 5.393974072220876, 6.033177115768474, 6.224664620334985, 6.982772284040608, 7.174143779364347, 7.697626654434459, 8.466361179978750, 8.876480915477667, 9.177229544924181, 9.908377885826377, 10.15313908233991, 10.73541847744766, 11.05560883852347, 11.76679452723744, 12.29671305945430, 12.65420979833123, 13.04383521920841
