| L(s) = 1 | − 3-s + 2·7-s + 9-s + 2·11-s + 2·13-s − 2·17-s + 8·19-s − 2·21-s − 5·25-s − 27-s − 6·29-s + 8·31-s − 2·33-s − 10·37-s − 2·39-s + 10·41-s − 4·43-s − 8·47-s − 3·49-s + 2·51-s − 12·53-s − 8·57-s + 12·59-s − 6·61-s + 2·63-s + 12·67-s − 12·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s − 0.485·17-s + 1.83·19-s − 0.436·21-s − 25-s − 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.348·33-s − 1.64·37-s − 0.320·39-s + 1.56·41-s − 0.609·43-s − 1.16·47-s − 3/7·49-s + 0.280·51-s − 1.64·53-s − 1.05·57-s + 1.56·59-s − 0.768·61-s + 0.251·63-s + 1.46·67-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 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
−13.86043630162051, −13.52856285599892, −13.14514467700594, −12.25397125230586, −12.02479568273377, −11.44741425668981, −11.21516366829186, −10.71705555818070, −9.977395174117877, −9.567135068150043, −9.167355342212650, −8.434331094357716, −7.909976970743293, −7.575251639052127, −6.797478835423934, −6.462917677038486, −5.777498789041669, −5.295165262140705, −4.835963286811486, −4.195376171939442, −3.608287284573605, −3.075845560080502, −2.089816904127357, −1.512251275243235, −0.9885629193813989, 0,
0.9885629193813989, 1.512251275243235, 2.089816904127357, 3.075845560080502, 3.608287284573605, 4.195376171939442, 4.835963286811486, 5.295165262140705, 5.777498789041669, 6.462917677038486, 6.797478835423934, 7.575251639052127, 7.909976970743293, 8.434331094357716, 9.167355342212650, 9.567135068150043, 9.977395174117877, 10.71705555818070, 11.21516366829186, 11.44741425668981, 12.02479568273377, 12.25397125230586, 13.14514467700594, 13.52856285599892, 13.86043630162051
