| L(s) = 1 | − 4·7-s − 4·11-s − 13-s − 4·17-s − 6·19-s − 5·25-s − 6·29-s + 4·31-s − 6·37-s + 6·41-s + 4·43-s − 8·47-s + 9·49-s + 6·53-s − 12·59-s − 10·61-s + 2·67-s − 12·71-s − 10·73-s + 16·77-s − 10·79-s − 4·83-s − 10·89-s + 4·91-s + 14·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1.20·11-s − 0.277·13-s − 0.970·17-s − 1.37·19-s − 25-s − 1.11·29-s + 0.718·31-s − 0.986·37-s + 0.937·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s + 0.824·53-s − 1.56·59-s − 1.28·61-s + 0.244·67-s − 1.42·71-s − 1.17·73-s + 1.82·77-s − 1.12·79-s − 0.439·83-s − 1.05·89-s + 0.419·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−16.53976848447123, −15.91394736880060, −15.60996781513513, −15.14094665265531, −14.42283529139380, −13.57891794856781, −13.19306522892756, −12.86461031282568, −12.28432933014841, −11.57539715790001, −10.76440263713947, −10.43090146084773, −9.816255279127684, −9.204065635246666, −8.693030456745556, −7.888330243609261, −7.317245230039509, −6.630171823741182, −6.058983807675200, −5.557342752941783, −4.547138812883162, −4.051835284654735, −3.098963866834441, −2.586790462252968, −1.776473230524683, 0, 0,
1.776473230524683, 2.586790462252968, 3.098963866834441, 4.051835284654735, 4.547138812883162, 5.557342752941783, 6.058983807675200, 6.630171823741182, 7.317245230039509, 7.888330243609261, 8.693030456745556, 9.204065635246666, 9.816255279127684, 10.43090146084773, 10.76440263713947, 11.57539715790001, 12.28432933014841, 12.86461031282568, 13.19306522892756, 13.57891794856781, 14.42283529139380, 15.14094665265531, 15.60996781513513, 15.91394736880060, 16.53976848447123
