| L(s) = 1 | − 3-s − 4·5-s + 9-s − 2·11-s − 13-s + 4·15-s + 6·17-s − 4·19-s − 4·23-s + 11·25-s − 27-s + 6·29-s + 8·31-s + 2·33-s + 10·37-s + 39-s + 4·41-s − 4·43-s − 4·45-s − 6·47-s − 6·51-s − 6·53-s + 8·55-s + 4·57-s + 6·59-s − 6·61-s + 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s + 1/3·9-s − 0.603·11-s − 0.277·13-s + 1.03·15-s + 1.45·17-s − 0.917·19-s − 0.834·23-s + 11/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.348·33-s + 1.64·37-s + 0.160·39-s + 0.624·41-s − 0.609·43-s − 0.596·45-s − 0.875·47-s − 0.840·51-s − 0.824·53-s + 1.07·55-s + 0.529·57-s + 0.781·59-s − 0.768·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 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
−13.77207659288552, −13.09389963947354, −12.70687461587409, −12.14145802771899, −11.89968563830622, −11.57039925735720, −10.82113491682054, −10.61601469079277, −9.898502184357397, −9.646204046611710, −8.607936518178839, −8.249198328984242, −7.839045613921547, −7.551988493044379, −6.850201098155303, −6.286166717362311, −5.860052027817102, −4.988142824758674, −4.620507883357153, −4.218520734307085, −3.541612477786888, −2.988441753696431, −2.418299764193012, −1.297570368147212, −0.6809079479069218, 0,
0.6809079479069218, 1.297570368147212, 2.418299764193012, 2.988441753696431, 3.541612477786888, 4.218520734307085, 4.620507883357153, 4.988142824758674, 5.860052027817102, 6.286166717362311, 6.850201098155303, 7.551988493044379, 7.839045613921547, 8.249198328984242, 8.607936518178839, 9.646204046611710, 9.898502184357397, 10.61601469079277, 10.82113491682054, 11.57039925735720, 11.89968563830622, 12.14145802771899, 12.70687461587409, 13.09389963947354, 13.77207659288552
