| L(s) = 1 | − 2·7-s − 3·9-s + 4·11-s − 2·13-s − 17-s + 4·19-s − 6·23-s − 5·25-s − 8·29-s + 2·31-s − 4·37-s − 2·41-s − 4·43-s − 12·47-s − 3·49-s + 6·53-s + 4·59-s − 4·61-s + 6·63-s + 4·67-s + 6·71-s − 6·73-s − 8·77-s + 10·79-s + 9·81-s − 12·83-s − 10·89-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 9-s + 1.20·11-s − 0.554·13-s − 0.242·17-s + 0.917·19-s − 1.25·23-s − 25-s − 1.48·29-s + 0.359·31-s − 0.657·37-s − 0.312·41-s − 0.609·43-s − 1.75·47-s − 3/7·49-s + 0.824·53-s + 0.520·59-s − 0.512·61-s + 0.755·63-s + 0.488·67-s + 0.712·71-s − 0.702·73-s − 0.911·77-s + 1.12·79-s + 81-s − 1.31·83-s − 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−9.594062577326757145430904108312, −8.711103350044857364339537245110, −7.83292879902090712076638540092, −6.84041393535243124859500516856, −6.09068663104670617751582887657, −5.28300507382393287491227047568, −3.95839388323488154494453195015, −3.20324136736330633519411958562, −1.88616587730267621143506081518, 0,
1.88616587730267621143506081518, 3.20324136736330633519411958562, 3.95839388323488154494453195015, 5.28300507382393287491227047568, 6.09068663104670617751582887657, 6.84041393535243124859500516856, 7.83292879902090712076638540092, 8.711103350044857364339537245110, 9.594062577326757145430904108312
