| L(s) = 1 | − 2-s − 3-s − 4-s + 6-s − 4·7-s + 3·8-s + 9-s + 4·11-s + 12-s − 6·13-s + 4·14-s − 16-s + 2·17-s − 18-s − 4·19-s + 4·21-s − 4·22-s + 23-s − 3·24-s + 6·26-s − 27-s + 4·28-s − 10·29-s − 8·31-s − 5·32-s − 4·33-s − 2·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.51·7-s + 1.06·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s − 1.66·13-s + 1.06·14-s − 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.917·19-s + 0.872·21-s − 0.852·22-s + 0.208·23-s − 0.612·24-s + 1.17·26-s − 0.192·27-s + 0.755·28-s − 1.85·29-s − 1.43·31-s − 0.883·32-s − 0.696·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3768624357\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3768624357\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 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.ac |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−9.248730845832821478964030694923, −9.011327379884043984081085396034, −7.50874407066058928772240206874, −7.15639099218325741612559400686, −6.17319109664919216573951817926, −5.34566445580048941613167398485, −4.26505050339368718721496501947, −3.53192868490368121303740772941, −2.01223314811043242019565240774, −0.47141820910232298718119603944,
0.47141820910232298718119603944, 2.01223314811043242019565240774, 3.53192868490368121303740772941, 4.26505050339368718721496501947, 5.34566445580048941613167398485, 6.17319109664919216573951817926, 7.15639099218325741612559400686, 7.50874407066058928772240206874, 9.011327379884043984081085396034, 9.248730845832821478964030694923
