| L(s) = 1 | + 2·5-s + 7-s + 2·13-s + 17-s − 5·19-s + 6·23-s − 25-s + 10·29-s + 9·31-s + 2·35-s − 11·37-s + 12·41-s − 4·43-s − 12·47-s − 6·49-s − 14·53-s + 10·59-s − 61-s + 4·65-s + 11·67-s − 7·73-s + 15·79-s + 2·83-s + 2·85-s + 8·89-s + 2·91-s − 10·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s + 0.554·13-s + 0.242·17-s − 1.14·19-s + 1.25·23-s − 1/5·25-s + 1.85·29-s + 1.61·31-s + 0.338·35-s − 1.80·37-s + 1.87·41-s − 0.609·43-s − 1.75·47-s − 6/7·49-s − 1.92·53-s + 1.30·59-s − 0.128·61-s + 0.496·65-s + 1.34·67-s − 0.819·73-s + 1.68·79-s + 0.219·83-s + 0.216·85-s + 0.847·89-s + 0.209·91-s − 1.02·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - 15 T + p T^{2} \) | 1.79.ap |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
| 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
−13.15383591792549, −12.41271355731574, −12.13491650821898, −11.48759239282142, −11.00474739389441, −10.67731770565181, −10.18746686260176, −9.661941509397573, −9.416258072257141, −8.611298885567295, −8.323496384201237, −8.092485653077579, −7.230325720013736, −6.694664336515718, −6.266479329722016, −6.112163153351422, −5.148792435284024, −4.931234409448035, −4.495718889352002, −3.689593143703568, −3.200395178738506, −2.559824044837870, −2.105722186167734, −1.353749474813527, −1.015900546073950, 0,
1.015900546073950, 1.353749474813527, 2.105722186167734, 2.559824044837870, 3.200395178738506, 3.689593143703568, 4.495718889352002, 4.931234409448035, 5.148792435284024, 6.112163153351422, 6.266479329722016, 6.694664336515718, 7.230325720013736, 8.092485653077579, 8.323496384201237, 8.611298885567295, 9.416258072257141, 9.661941509397573, 10.18746686260176, 10.67731770565181, 11.00474739389441, 11.48759239282142, 12.13491650821898, 12.41271355731574, 13.15383591792549
