| L(s) = 1 | − 2·5-s − 11-s − 2·13-s + 6·17-s − 8·23-s − 25-s + 6·29-s + 4·31-s + 2·37-s + 6·41-s + 8·43-s + 8·47-s + 2·53-s + 2·55-s − 4·59-s − 2·61-s + 4·65-s + 12·67-s − 10·73-s − 4·83-s − 12·85-s − 14·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.301·11-s − 0.554·13-s + 1.45·17-s − 1.66·23-s − 1/5·25-s + 1.11·29-s + 0.718·31-s + 0.328·37-s + 0.937·41-s + 1.21·43-s + 1.16·47-s + 0.274·53-s + 0.269·55-s − 0.520·59-s − 0.256·61-s + 0.496·65-s + 1.46·67-s − 1.17·73-s − 0.439·83-s − 1.30·85-s − 1.48·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−12.61010275487770, −12.34973012823858, −12.01098283014120, −11.71194970082213, −11.05405017027416, −10.57763519290665, −10.17257674923558, −9.654460091384092, −9.406770832105557, −8.524509136302781, −8.169287442107451, −7.849327226944456, −7.411522777272500, −6.985254755839404, −6.251170062190531, −5.745052559827317, −5.485690110111453, −4.640358627950687, −4.234299396613116, −3.901534045485275, −3.181491031659268, −2.676471343794644, −2.198164730327936, −1.283134404941428, −0.7295751833652960, 0,
0.7295751833652960, 1.283134404941428, 2.198164730327936, 2.676471343794644, 3.181491031659268, 3.901534045485275, 4.234299396613116, 4.640358627950687, 5.485690110111453, 5.745052559827317, 6.251170062190531, 6.985254755839404, 7.411522777272500, 7.849327226944456, 8.169287442107451, 8.524509136302781, 9.406770832105557, 9.654460091384092, 10.17257674923558, 10.57763519290665, 11.05405017027416, 11.71194970082213, 12.01098283014120, 12.34973012823858, 12.61010275487770
