| L(s) = 1 | − 3-s + 2·7-s + 9-s − 4·13-s − 3·17-s + 4·19-s − 2·21-s + 3·23-s − 27-s − 31-s − 2·37-s + 4·39-s − 6·41-s + 8·43-s − 3·47-s − 3·49-s + 3·51-s − 9·53-s − 4·57-s + 12·59-s − 5·61-s + 2·63-s − 2·67-s − 3·69-s + 12·71-s + 8·73-s + 79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s − 1.10·13-s − 0.727·17-s + 0.917·19-s − 0.436·21-s + 0.625·23-s − 0.192·27-s − 0.179·31-s − 0.328·37-s + 0.640·39-s − 0.937·41-s + 1.21·43-s − 0.437·47-s − 3/7·49-s + 0.420·51-s − 1.23·53-s − 0.529·57-s + 1.56·59-s − 0.640·61-s + 0.251·63-s − 0.244·67-s − 0.361·69-s + 1.42·71-s + 0.936·73-s + 0.112·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
| 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
−15.12715531106290, −14.67020947728017, −14.16286019787867, −13.62501131526344, −12.99391911967560, −12.48428663083170, −11.95686170058401, −11.48080221252119, −11.00483417801024, −10.54084923839519, −9.811022724537407, −9.398051060424028, −8.805045419880823, −7.972259677806400, −7.686998694192412, −6.871604187294406, −6.620759049471590, −5.660597926184961, −5.102540253739967, −4.825375183227210, −4.081236461851218, −3.281686307869502, −2.476052791081971, −1.792032187797264, −0.9665939077382219, 0,
0.9665939077382219, 1.792032187797264, 2.476052791081971, 3.281686307869502, 4.081236461851218, 4.825375183227210, 5.102540253739967, 5.660597926184961, 6.620759049471590, 6.871604187294406, 7.686998694192412, 7.972259677806400, 8.805045419880823, 9.398051060424028, 9.811022724537407, 10.54084923839519, 11.00483417801024, 11.48080221252119, 11.95686170058401, 12.48428663083170, 12.99391911967560, 13.62501131526344, 14.16286019787867, 14.67020947728017, 15.12715531106290
