| L(s) = 1 | − 2-s + 4-s − 5-s − 3·7-s − 8-s + 10-s − 11-s + 3·14-s + 16-s − 2·17-s − 2·19-s − 20-s + 22-s − 3·23-s + 25-s − 3·28-s − 29-s − 7·31-s − 32-s + 2·34-s + 3·35-s + 2·37-s + 2·38-s + 40-s + 5·41-s + 11·43-s − 44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.13·7-s − 0.353·8-s + 0.316·10-s − 0.301·11-s + 0.801·14-s + 1/4·16-s − 0.485·17-s − 0.458·19-s − 0.223·20-s + 0.213·22-s − 0.625·23-s + 1/5·25-s − 0.566·28-s − 0.185·29-s − 1.25·31-s − 0.176·32-s + 0.342·34-s + 0.507·35-s + 0.328·37-s + 0.324·38-s + 0.158·40-s + 0.780·41-s + 1.67·43-s − 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 167310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 - 11 T + p T^{2} \) | 1.47.al |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.57116539007856, −13.14785459627139, −12.69527044786663, −12.18349487811553, −11.99921716497429, −10.99946393765791, −10.86766959253837, −10.52634971297561, −9.761651607486417, −9.274872848738009, −9.145833470561316, −8.491408930186791, −7.825804683934222, −7.504100941044247, −7.055173964251109, −6.431765845667323, −5.856076737652396, −5.730751387772781, −4.577993330050674, −4.278227978179496, −3.577992414965081, −3.002536844199132, −2.524282832153559, −1.837313146999062, −1.055428487996775, 0, 0,
1.055428487996775, 1.837313146999062, 2.524282832153559, 3.002536844199132, 3.577992414965081, 4.278227978179496, 4.577993330050674, 5.730751387772781, 5.856076737652396, 6.431765845667323, 7.055173964251109, 7.504100941044247, 7.825804683934222, 8.491408930186791, 9.145833470561316, 9.274872848738009, 9.761651607486417, 10.52634971297561, 10.86766959253837, 10.99946393765791, 11.99921716497429, 12.18349487811553, 12.69527044786663, 13.14785459627139, 13.57116539007856
