| L(s) = 1 | + 2-s + 4-s − 5-s + 4·7-s + 8-s − 10-s + 4·14-s + 16-s − 17-s + 4·19-s − 20-s + 25-s + 4·28-s − 6·29-s + 4·31-s + 32-s − 34-s − 4·35-s − 2·37-s + 4·38-s − 40-s + 6·41-s − 4·43-s + 9·49-s + 50-s − 6·53-s + 4·56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s + 0.353·8-s − 0.316·10-s + 1.06·14-s + 1/4·16-s − 0.242·17-s + 0.917·19-s − 0.223·20-s + 1/5·25-s + 0.755·28-s − 1.11·29-s + 0.718·31-s + 0.176·32-s − 0.171·34-s − 0.676·35-s − 0.328·37-s + 0.648·38-s − 0.158·40-s + 0.937·41-s − 0.609·43-s + 9/7·49-s + 0.141·50-s − 0.824·53-s + 0.534·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.11532769320901, −12.47110297611573, −12.15461605836817, −11.61587104860052, −11.31124577263240, −10.95527573919618, −10.51347496331560, −9.849775117333903, −9.320782036540993, −8.808260971649731, −8.141423746227409, −7.955477172646313, −7.321667003893912, −7.107465545859935, −6.312552924642114, −5.753861506110184, −5.377528216537951, −4.757030151164860, −4.430736808701661, −4.025101717689329, −3.147209397221988, −2.944073811549433, −1.983972371777869, −1.609262749466985, −0.9938080664820121, 0,
0.9938080664820121, 1.609262749466985, 1.983972371777869, 2.944073811549433, 3.147209397221988, 4.025101717689329, 4.430736808701661, 4.757030151164860, 5.377528216537951, 5.753861506110184, 6.312552924642114, 7.107465545859935, 7.321667003893912, 7.955477172646313, 8.141423746227409, 8.808260971649731, 9.320782036540993, 9.849775117333903, 10.51347496331560, 10.95527573919618, 11.31124577263240, 11.61587104860052, 12.15461605836817, 12.47110297611573, 13.11532769320901
