| L(s) = 1 | − 3-s + 2·7-s + 9-s + 6·11-s − 2·13-s + 17-s − 19-s − 2·21-s + 2·23-s − 27-s − 8·29-s + 4·31-s − 6·33-s − 8·37-s + 2·39-s − 8·41-s − 4·43-s + 4·47-s − 3·49-s − 51-s + 6·53-s + 57-s + 4·59-s − 10·61-s + 2·63-s + 4·67-s − 2·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s + 1.80·11-s − 0.554·13-s + 0.242·17-s − 0.229·19-s − 0.436·21-s + 0.417·23-s − 0.192·27-s − 1.48·29-s + 0.718·31-s − 1.04·33-s − 1.31·37-s + 0.320·39-s − 1.24·41-s − 0.609·43-s + 0.583·47-s − 3/7·49-s − 0.140·51-s + 0.824·53-s + 0.132·57-s + 0.520·59-s − 1.28·61-s + 0.251·63-s + 0.488·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−12.35829524613770, −12.16023103621670, −11.89707249947658, −11.28121189799107, −11.05643360333553, −10.52298511457336, −9.954039174328671, −9.496105625605749, −9.194542122672561, −8.474340061869764, −8.309632999513235, −7.532980101253859, −7.089312658344295, −6.714169851204780, −6.330631113328780, −5.610725685410028, −5.273052829956374, −4.754141659111209, −4.272927211855931, −3.693922687633007, −3.379825943730785, −2.459475253109144, −1.769971629551863, −1.489245289458024, −0.8017484082020205, 0,
0.8017484082020205, 1.489245289458024, 1.769971629551863, 2.459475253109144, 3.379825943730785, 3.693922687633007, 4.272927211855931, 4.754141659111209, 5.273052829956374, 5.610725685410028, 6.330631113328780, 6.714169851204780, 7.089312658344295, 7.532980101253859, 8.309632999513235, 8.474340061869764, 9.194542122672561, 9.496105625605749, 9.954039174328671, 10.52298511457336, 11.05643360333553, 11.28121189799107, 11.89707249947658, 12.16023103621670, 12.35829524613770
