| L(s) = 1 | − 5-s + 7-s + 4·11-s − 2·13-s − 2·17-s − 4·19-s + 25-s + 2·29-s − 8·31-s − 35-s − 2·37-s − 2·41-s − 4·43-s + 49-s + 10·53-s − 4·55-s − 12·59-s + 6·61-s + 2·65-s − 12·67-s − 6·73-s + 4·77-s + 8·79-s + 4·83-s + 2·85-s − 2·89-s − 2·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 1.20·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s + 1/5·25-s + 0.371·29-s − 1.43·31-s − 0.169·35-s − 0.328·37-s − 0.312·41-s − 0.609·43-s + 1/7·49-s + 1.37·53-s − 0.539·55-s − 1.56·59-s + 0.768·61-s + 0.248·65-s − 1.46·67-s − 0.702·73-s + 0.455·77-s + 0.900·79-s + 0.439·83-s + 0.216·85-s − 0.211·89-s − 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 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 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
| 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
−7.86409298701159649329849414300, −7.10614628230433984877857849164, −6.58459022828675244730732149364, −5.71468836174001427851387787891, −4.79045184918107241481368991566, −4.16780241437033380210574539654, −3.44532387481355093858055635099, −2.30537352233446591345492900960, −1.41038467674394923031748304403, 0,
1.41038467674394923031748304403, 2.30537352233446591345492900960, 3.44532387481355093858055635099, 4.16780241437033380210574539654, 4.79045184918107241481368991566, 5.71468836174001427851387787891, 6.58459022828675244730732149364, 7.10614628230433984877857849164, 7.86409298701159649329849414300
