| L(s) = 1 | − 2·4-s + 7-s + 5·11-s + 4·16-s − 7·17-s − 8·19-s − 6·23-s − 2·28-s + 5·29-s − 2·31-s − 6·37-s − 12·41-s + 2·43-s − 10·44-s − 47-s + 49-s + 2·53-s − 8·59-s − 2·61-s − 8·64-s − 10·67-s + 14·68-s + 5·71-s + 11·73-s + 16·76-s + 5·77-s − 8·79-s + ⋯ |
| L(s) = 1 | − 4-s + 0.377·7-s + 1.50·11-s + 16-s − 1.69·17-s − 1.83·19-s − 1.25·23-s − 0.377·28-s + 0.928·29-s − 0.359·31-s − 0.986·37-s − 1.87·41-s + 0.304·43-s − 1.50·44-s − 0.145·47-s + 1/7·49-s + 0.274·53-s − 1.04·59-s − 0.256·61-s − 64-s − 1.22·67-s + 1.69·68-s + 0.593·71-s + 1.28·73-s + 1.83·76-s + 0.569·77-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 266175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−13.42527902436282, −12.69541054573413, −12.36892458812328, −11.99091934524081, −11.43388323484851, −10.90804827814064, −10.48192398086575, −10.02521951151284, −9.475619087882348, −8.880308675428411, −8.731890256268640, −8.333379555368166, −7.825166712629088, −6.987661924579134, −6.564039544056256, −6.309964480797901, −5.647748784294174, −4.946556737430776, −4.533252000800445, −4.051426720072456, −3.872363151824100, −3.066070376144725, −2.206293098330392, −1.755493037388824, −1.194062467441160, 0, 0,
1.194062467441160, 1.755493037388824, 2.206293098330392, 3.066070376144725, 3.872363151824100, 4.051426720072456, 4.533252000800445, 4.946556737430776, 5.647748784294174, 6.309964480797901, 6.564039544056256, 6.987661924579134, 7.825166712629088, 8.333379555368166, 8.731890256268640, 8.880308675428411, 9.475619087882348, 10.02521951151284, 10.48192398086575, 10.90804827814064, 11.43388323484851, 11.99091934524081, 12.36892458812328, 12.69541054573413, 13.42527902436282
