| L(s) = 1 | − 4·5-s − 2·7-s + 6·13-s + 17-s − 4·19-s − 6·23-s + 11·25-s − 4·29-s − 6·31-s + 8·35-s + 4·37-s + 10·41-s + 4·43-s − 4·47-s − 3·49-s − 2·53-s + 12·59-s + 4·61-s − 24·65-s + 12·67-s + 6·71-s + 2·73-s + 10·79-s − 12·83-s − 4·85-s + 2·89-s − 12·91-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.755·7-s + 1.66·13-s + 0.242·17-s − 0.917·19-s − 1.25·23-s + 11/5·25-s − 0.742·29-s − 1.07·31-s + 1.35·35-s + 0.657·37-s + 1.56·41-s + 0.609·43-s − 0.583·47-s − 3/7·49-s − 0.274·53-s + 1.56·59-s + 0.512·61-s − 2.97·65-s + 1.46·67-s + 0.712·71-s + 0.234·73-s + 1.12·79-s − 1.31·83-s − 0.433·85-s + 0.211·89-s − 1.25·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−7.49560306106886736233413977331, −6.60015236239644657099609663290, −6.15763306192443026020969574323, −5.28719191878192357874088442322, −4.10377059400313729374689043670, −3.91091385017041077335507391788, −3.35019242624604968600631282015, −2.25344849290415503147337277629, −0.942173780326292969142276990846, 0,
0.942173780326292969142276990846, 2.25344849290415503147337277629, 3.35019242624604968600631282015, 3.91091385017041077335507391788, 4.10377059400313729374689043670, 5.28719191878192357874088442322, 6.15763306192443026020969574323, 6.60015236239644657099609663290, 7.49560306106886736233413977331
