| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 12-s − 13-s + 16-s + 2·17-s − 18-s + 4·19-s + 24-s + 26-s − 27-s + 6·29-s − 4·31-s − 32-s − 2·34-s + 36-s − 2·37-s − 4·38-s + 39-s − 6·41-s + 12·47-s − 48-s − 7·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.288·12-s − 0.277·13-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.204·24-s + 0.196·26-s − 0.192·27-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.342·34-s + 1/6·36-s − 0.328·37-s − 0.648·38-s + 0.160·39-s − 0.937·41-s + 1.75·47-s − 0.144·48-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8080211595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8080211595\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−12.75316203373731, −12.31927770826153, −11.80632043800223, −11.63189903599059, −11.00300702179160, −10.46135538759754, −10.14809752016963, −9.792047767358828, −9.078663438366822, −8.838719337162829, −8.195114191848943, −7.646298919048027, −7.199328276795278, −6.951223111010271, −6.108996314444867, −5.856739951063005, −5.291945699438314, −4.692381724944221, −4.246279669291642, −3.341463444835233, −3.061118391791803, −2.312208429140190, −1.528552411593940, −1.156967440258089, −0.3121538810961457,
0.3121538810961457, 1.156967440258089, 1.528552411593940, 2.312208429140190, 3.061118391791803, 3.341463444835233, 4.246279669291642, 4.692381724944221, 5.291945699438314, 5.856739951063005, 6.108996314444867, 6.951223111010271, 7.199328276795278, 7.646298919048027, 8.195114191848943, 8.838719337162829, 9.078663438366822, 9.792047767358828, 10.14809752016963, 10.46135538759754, 11.00300702179160, 11.63189903599059, 11.80632043800223, 12.31927770826153, 12.75316203373731
