| L(s) = 1 | + 2-s + 4-s + 5-s + 2·7-s + 8-s + 10-s − 13-s + 2·14-s + 16-s − 2·17-s + 20-s − 8·23-s + 25-s − 26-s + 2·28-s − 8·29-s + 6·31-s + 32-s − 2·34-s + 2·35-s + 40-s + 10·41-s + 8·43-s − 8·46-s − 3·49-s + 50-s − 52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s + 0.353·8-s + 0.316·10-s − 0.277·13-s + 0.534·14-s + 1/4·16-s − 0.485·17-s + 0.223·20-s − 1.66·23-s + 1/5·25-s − 0.196·26-s + 0.377·28-s − 1.48·29-s + 1.07·31-s + 0.176·32-s − 0.342·34-s + 0.338·35-s + 0.158·40-s + 1.56·41-s + 1.21·43-s − 1.17·46-s − 3/7·49-s + 0.141·50-s − 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
| 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
−13.71105133669659, −13.21223508966390, −12.61217305481476, −12.40477722713645, −11.61792490334502, −11.38014188290692, −10.91734781692076, −10.32174578864755, −9.801281764479111, −9.447222080224227, −8.683692281337653, −8.247564196811125, −7.699522645639093, −7.259994157689201, −6.667457841545279, −6.048010638721989, −5.634522515905066, −5.265789172722952, −4.381783539126689, −4.238594035283658, −3.622462672586791, −2.623394967963376, −2.383207963739873, −1.718491888523806, −1.049201423291567, 0,
1.049201423291567, 1.718491888523806, 2.383207963739873, 2.623394967963376, 3.622462672586791, 4.238594035283658, 4.381783539126689, 5.265789172722952, 5.634522515905066, 6.048010638721989, 6.667457841545279, 7.259994157689201, 7.699522645639093, 8.247564196811125, 8.683692281337653, 9.447222080224227, 9.801281764479111, 10.32174578864755, 10.91734781692076, 11.38014188290692, 11.61792490334502, 12.40477722713645, 12.61217305481476, 13.21223508966390, 13.71105133669659
