| L(s) = 1 | − 2·5-s + 4·7-s + 2·13-s − 2·17-s − 23-s − 25-s − 2·29-s − 8·35-s − 10·37-s − 6·41-s − 8·43-s + 8·47-s + 9·49-s + 6·53-s + 4·59-s − 14·61-s − 4·65-s + 8·67-s + 8·71-s + 6·73-s − 12·79-s − 12·83-s + 4·85-s + 2·89-s + 8·91-s + 10·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s + 0.554·13-s − 0.485·17-s − 0.208·23-s − 1/5·25-s − 0.371·29-s − 1.35·35-s − 1.64·37-s − 0.937·41-s − 1.21·43-s + 1.16·47-s + 9/7·49-s + 0.824·53-s + 0.520·59-s − 1.79·61-s − 0.496·65-s + 0.977·67-s + 0.949·71-s + 0.702·73-s − 1.35·79-s − 1.31·83-s + 0.433·85-s + 0.211·89-s + 0.838·91-s + 1.01·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.752126878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.752126878\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 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.ak |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.14365656048216, −12.33523573081618, −12.13012694510525, −11.53868951220158, −11.28606517037464, −10.89185994685751, −10.29090283332743, −9.934795422906104, −9.002364039666773, −8.722839972353804, −8.277927979040893, −7.938151529599118, −7.272976985838575, −7.057347235733811, −6.306742623057956, −5.675331576646378, −5.181977732609552, −4.686580541974506, −4.222674517787585, −3.642913103590413, −3.262732750747147, −2.243518568816602, −1.848419840149222, −1.225852467417351, −0.3844008005240905,
0.3844008005240905, 1.225852467417351, 1.848419840149222, 2.243518568816602, 3.262732750747147, 3.642913103590413, 4.222674517787585, 4.686580541974506, 5.181977732609552, 5.675331576646378, 6.306742623057956, 7.057347235733811, 7.272976985838575, 7.938151529599118, 8.277927979040893, 8.722839972353804, 9.002364039666773, 9.934795422906104, 10.29090283332743, 10.89185994685751, 11.28606517037464, 11.53868951220158, 12.13012694510525, 12.33523573081618, 13.14365656048216
