| L(s) = 1 | − 3-s + 9-s − 6·11-s − 13-s + 2·17-s − 4·23-s − 5·25-s − 27-s − 6·29-s + 4·31-s + 6·33-s − 2·37-s + 39-s − 4·43-s − 10·47-s − 7·49-s − 2·51-s − 10·53-s + 6·59-s − 6·61-s + 12·67-s + 4·69-s − 2·71-s + 6·73-s + 5·75-s + 16·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.80·11-s − 0.277·13-s + 0.485·17-s − 0.834·23-s − 25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 1.04·33-s − 0.328·37-s + 0.160·39-s − 0.609·43-s − 1.45·47-s − 49-s − 0.280·51-s − 1.37·53-s + 0.781·59-s − 0.768·61-s + 1.46·67-s + 0.481·69-s − 0.237·71-s + 0.702·73-s + 0.577·75-s + 1.80·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−10.15630188499991457516970890674, −9.623331085660540106579040249977, −8.109170288766974377318628852457, −7.70721800445715600502657637543, −6.46852629489327197023949075773, −5.50835748283028813340348869563, −4.80983646216625589523400190905, −3.41767268718365275694731336535, −2.05225824663803178760552738615, 0,
2.05225824663803178760552738615, 3.41767268718365275694731336535, 4.80983646216625589523400190905, 5.50835748283028813340348869563, 6.46852629489327197023949075773, 7.70721800445715600502657637543, 8.109170288766974377318628852457, 9.623331085660540106579040249977, 10.15630188499991457516970890674
