| L(s) = 1 | + 3-s − 2·4-s − 3·5-s − 7-s + 9-s − 2·12-s + 4·13-s − 3·15-s + 4·16-s + 3·17-s − 2·19-s + 6·20-s − 21-s + 4·25-s + 27-s + 2·28-s + 6·29-s + 2·31-s + 3·35-s − 2·36-s − 10·37-s + 4·39-s − 6·41-s − 11·43-s − 3·45-s − 3·47-s + 4·48-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 1.34·5-s − 0.377·7-s + 1/3·9-s − 0.577·12-s + 1.10·13-s − 0.774·15-s + 16-s + 0.727·17-s − 0.458·19-s + 1.34·20-s − 0.218·21-s + 4/5·25-s + 0.192·27-s + 0.377·28-s + 1.11·29-s + 0.359·31-s + 0.507·35-s − 1/3·36-s − 1.64·37-s + 0.640·39-s − 0.937·41-s − 1.67·43-s − 0.447·45-s − 0.437·47-s + 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
| 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
−8.593565044716048297127328972129, −8.005581478759224187717954077599, −7.18370532094215395583297187649, −6.27519249095712814961967521422, −5.16378096587245035333983781754, −4.32740310692458104162463103471, −3.61284231591070088832057974545, −3.13539084113908463199933527201, −1.32939120782437545390216757653, 0,
1.32939120782437545390216757653, 3.13539084113908463199933527201, 3.61284231591070088832057974545, 4.32740310692458104162463103471, 5.16378096587245035333983781754, 6.27519249095712814961967521422, 7.18370532094215395583297187649, 8.005581478759224187717954077599, 8.593565044716048297127328972129
