| L(s) = 1 | − 5-s − 2·7-s − 3·11-s − 3·13-s − 17-s + 2·19-s − 23-s + 25-s + 9·31-s + 2·35-s + 12·37-s − 41-s − 43-s − 8·47-s − 3·49-s − 5·53-s + 3·55-s + 12·59-s + 12·61-s + 3·65-s + 13·67-s + 6·71-s − 8·73-s + 6·77-s − 8·79-s − 3·83-s + 85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 0.904·11-s − 0.832·13-s − 0.242·17-s + 0.458·19-s − 0.208·23-s + 1/5·25-s + 1.61·31-s + 0.338·35-s + 1.97·37-s − 0.156·41-s − 0.152·43-s − 1.16·47-s − 3/7·49-s − 0.686·53-s + 0.404·55-s + 1.56·59-s + 1.53·61-s + 0.372·65-s + 1.58·67-s + 0.712·71-s − 0.936·73-s + 0.683·77-s − 0.900·79-s − 0.329·83-s + 0.108·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.198598975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.198598975\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 43 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 - 12 T + p T^{2} \) | 1.37.am |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
| 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
−13.25349781282869, −13.00643130042438, −12.81865773987406, −11.98258065877784, −11.62257240669643, −11.28228886144536, −10.52879187463176, −10.02223639543641, −9.740711603354464, −9.324665492049517, −8.390944977368440, −8.194307889088520, −7.637840561340855, −7.116284599271642, −6.489675518167059, −6.213774136748789, −5.272688851890477, −5.055418588685986, −4.329706241639593, −3.824070518522740, −3.032924296383038, −2.696215110616059, −2.114543100729303, −1.035139354527333, −0.3773613557128663,
0.3773613557128663, 1.035139354527333, 2.114543100729303, 2.696215110616059, 3.032924296383038, 3.824070518522740, 4.329706241639593, 5.055418588685986, 5.272688851890477, 6.213774136748789, 6.489675518167059, 7.116284599271642, 7.637840561340855, 8.194307889088520, 8.390944977368440, 9.324665492049517, 9.740711603354464, 10.02223639543641, 10.52879187463176, 11.28228886144536, 11.62257240669643, 11.98258065877784, 12.81865773987406, 13.00643130042438, 13.25349781282869
