| L(s) = 1 | − 3·9-s + 2·13-s − 4·19-s − 5·25-s − 8·29-s − 31-s + 8·37-s + 4·43-s − 7·49-s − 6·53-s − 4·59-s − 8·61-s − 12·67-s − 8·73-s + 9·81-s − 4·83-s + 6·89-s − 16·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 6·117-s + ⋯ |
| L(s) = 1 | − 9-s + 0.554·13-s − 0.917·19-s − 25-s − 1.48·29-s − 0.179·31-s + 1.31·37-s + 0.609·43-s − 49-s − 0.824·53-s − 0.520·59-s − 1.02·61-s − 1.46·67-s − 0.936·73-s + 81-s − 0.439·83-s + 0.635·89-s − 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.554·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2525491077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2525491077\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 31 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−13.49589994325205, −13.04136084721526, −12.23290737873389, −12.12857487819795, −11.26225795748516, −11.05075764541572, −10.75978481613495, −9.955525956366972, −9.362890295478157, −9.182125855805229, −8.474532070273474, −8.016809898279699, −7.675888548391273, −7.004284379886023, −6.215496252892266, −6.072442911788729, −5.519629748727012, −4.913373088702466, −4.087796225166281, −3.933051607253168, −2.992315722276767, −2.683010830894890, −1.823263869040315, −1.341157017295761, −0.1442419690545721,
0.1442419690545721, 1.341157017295761, 1.823263869040315, 2.683010830894890, 2.992315722276767, 3.933051607253168, 4.087796225166281, 4.913373088702466, 5.519629748727012, 6.072442911788729, 6.215496252892266, 7.004284379886023, 7.675888548391273, 8.016809898279699, 8.474532070273474, 9.182125855805229, 9.362890295478157, 9.955525956366972, 10.75978481613495, 11.05075764541572, 11.26225795748516, 12.12857487819795, 12.23290737873389, 13.04136084721526, 13.49589994325205
