| L(s) = 1 | − 3-s + 5-s − 7-s − 2·9-s + 2·13-s − 15-s − 6·17-s + 3·19-s + 21-s + 6·23-s − 4·25-s + 5·27-s + 3·29-s + 4·31-s − 35-s + 2·37-s − 2·39-s − 5·41-s − 2·45-s − 2·47-s − 6·49-s + 6·51-s − 3·53-s − 3·57-s + 59-s − 12·61-s + 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s − 2/3·9-s + 0.554·13-s − 0.258·15-s − 1.45·17-s + 0.688·19-s + 0.218·21-s + 1.25·23-s − 4/5·25-s + 0.962·27-s + 0.557·29-s + 0.718·31-s − 0.169·35-s + 0.328·37-s − 0.320·39-s − 0.780·41-s − 0.298·45-s − 0.291·47-s − 6/7·49-s + 0.840·51-s − 0.412·53-s − 0.397·57-s + 0.130·59-s − 1.53·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 59 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 15 T + p T^{2} \) | 1.79.p |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−8.289492059578755277930037991193, −7.23503296602818369515270586627, −6.43861163420690481107923289495, −6.05909468696688153232644342575, −5.15803147848862332612722517294, −4.49694535208075381432652481944, −3.31473196076294193224612457762, −2.57873205237227455984509773778, −1.32843621990161482531558900839, 0,
1.32843621990161482531558900839, 2.57873205237227455984509773778, 3.31473196076294193224612457762, 4.49694535208075381432652481944, 5.15803147848862332612722517294, 6.05909468696688153232644342575, 6.43861163420690481107923289495, 7.23503296602818369515270586627, 8.289492059578755277930037991193
