| L(s) = 1 | − 3·5-s − 3·11-s + 13-s − 7·17-s + 19-s − 23-s + 4·25-s − 2·29-s + 9·31-s + 3·37-s − 10·41-s − 4·43-s − 3·47-s − 53-s + 9·55-s − 11·59-s + 61-s − 3·65-s + 7·67-s − 8·71-s − 7·73-s − 11·79-s − 4·83-s + 21·85-s − 89-s − 3·95-s + 2·97-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.904·11-s + 0.277·13-s − 1.69·17-s + 0.229·19-s − 0.208·23-s + 4/5·25-s − 0.371·29-s + 1.61·31-s + 0.493·37-s − 1.56·41-s − 0.609·43-s − 0.437·47-s − 0.137·53-s + 1.21·55-s − 1.43·59-s + 0.128·61-s − 0.372·65-s + 0.855·67-s − 0.949·71-s − 0.819·73-s − 1.23·79-s − 0.439·83-s + 2.27·85-s − 0.105·89-s − 0.307·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 366912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + 11 T + p T^{2} \) | 1.59.l |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 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
−12.79743482747857, −12.17034010291879, −11.64464282051550, −11.53790731071722, −11.02229170568073, −10.54154742325340, −10.06765873930329, −9.665497323263322, −8.854533400481422, −8.555828435159155, −8.245660398282465, −7.700723756288172, −7.258563306653858, −6.878150954844761, −6.179423197520230, −5.943041203730555, −4.945256263339454, −4.728532676125892, −4.352805769909178, −3.647864311756290, −3.241120000984169, −2.682647864432714, −2.079087134667207, −1.394104768746960, −0.4954316545192265, 0,
0.4954316545192265, 1.394104768746960, 2.079087134667207, 2.682647864432714, 3.241120000984169, 3.647864311756290, 4.352805769909178, 4.728532676125892, 4.945256263339454, 5.943041203730555, 6.179423197520230, 6.878150954844761, 7.258563306653858, 7.700723756288172, 8.245660398282465, 8.555828435159155, 8.854533400481422, 9.665497323263322, 10.06765873930329, 10.54154742325340, 11.02229170568073, 11.53790731071722, 11.64464282051550, 12.17034010291879, 12.79743482747857
