| L(s) = 1 | − 2-s + 4-s − 5-s − 2·7-s − 8-s + 10-s + 5·13-s + 2·14-s + 16-s + 17-s + 6·19-s − 20-s − 6·23-s − 4·25-s − 5·26-s − 2·28-s + 5·29-s − 31-s − 32-s − 34-s + 2·35-s + 37-s − 6·38-s + 40-s − 9·41-s + 4·43-s + 6·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s − 0.353·8-s + 0.316·10-s + 1.38·13-s + 0.534·14-s + 1/4·16-s + 0.242·17-s + 1.37·19-s − 0.223·20-s − 1.25·23-s − 4/5·25-s − 0.980·26-s − 0.377·28-s + 0.928·29-s − 0.179·31-s − 0.176·32-s − 0.171·34-s + 0.338·35-s + 0.164·37-s − 0.973·38-s + 0.158·40-s − 1.40·41-s + 0.609·43-s + 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67518 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67518 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 11 | \( 1 \) | |
| 31 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 + 9 T + p T^{2} \) | 1.97.j |
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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
−14.27602676130798, −13.97285419308251, −13.44978679040062, −12.92933584584752, −12.19067472876771, −11.94239712538742, −11.32982586676471, −10.97672664074634, −10.17890749910858, −9.851526074042678, −9.493317038399973, −8.677426886209602, −8.329858121009015, −7.857112593026686, −7.246242643101099, −6.704388993103794, −6.093378427180805, −5.758214901970017, −4.976499407679032, −4.119330109511241, −3.494324363203803, −3.244163758102956, −2.321845895397971, −1.521908814047104, −0.8494925215699103, 0,
0.8494925215699103, 1.521908814047104, 2.321845895397971, 3.244163758102956, 3.494324363203803, 4.119330109511241, 4.976499407679032, 5.758214901970017, 6.093378427180805, 6.704388993103794, 7.246242643101099, 7.857112593026686, 8.329858121009015, 8.677426886209602, 9.493317038399973, 9.851526074042678, 10.17890749910858, 10.97672664074634, 11.32982586676471, 11.94239712538742, 12.19067472876771, 12.92933584584752, 13.44978679040062, 13.97285419308251, 14.27602676130798
