| L(s) = 1 | − 3-s − 5-s + 3·7-s + 9-s − 5·11-s − 2·13-s + 15-s − 17-s + 5·19-s − 3·21-s + 6·23-s + 25-s − 27-s − 3·29-s − 2·31-s + 5·33-s − 3·35-s − 7·37-s + 2·39-s + 41-s + 4·43-s − 45-s − 3·47-s + 2·49-s + 51-s − 9·53-s + 5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s − 1.50·11-s − 0.554·13-s + 0.258·15-s − 0.242·17-s + 1.14·19-s − 0.654·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s − 0.557·29-s − 0.359·31-s + 0.870·33-s − 0.507·35-s − 1.15·37-s + 0.320·39-s + 0.156·41-s + 0.609·43-s − 0.149·45-s − 0.437·47-s + 2/7·49-s + 0.140·51-s − 1.23·53-s + 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - T + p T^{2} \) | 1.41.ab |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
| 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
−7.82019860053062731851275989476, −7.51746771871297983551565905362, −6.74776346033095858563796970806, −5.47042446491171288475524183444, −5.16631113669803587358946354338, −4.54837901154406062612863330747, −3.39129618438877851688865417591, −2.44340682302722181835961385999, −1.31243317742938213545253608429, 0,
1.31243317742938213545253608429, 2.44340682302722181835961385999, 3.39129618438877851688865417591, 4.54837901154406062612863330747, 5.16631113669803587358946354338, 5.47042446491171288475524183444, 6.74776346033095858563796970806, 7.51746771871297983551565905362, 7.82019860053062731851275989476
