| L(s) = 1 | − 7-s + 5·11-s − 4·13-s − 4·17-s − 8·19-s − 8·23-s − 2·29-s + 7·31-s − 8·37-s + 4·43-s + 4·47-s − 6·49-s − 7·53-s + 12·67-s + 4·71-s + 15·73-s − 5·77-s + 4·79-s − 83-s + 12·89-s + 4·91-s − 7·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.50·11-s − 1.10·13-s − 0.970·17-s − 1.83·19-s − 1.66·23-s − 0.371·29-s + 1.25·31-s − 1.31·37-s + 0.609·43-s + 0.583·47-s − 6/7·49-s − 0.961·53-s + 1.46·67-s + 0.474·71-s + 1.75·73-s − 0.569·77-s + 0.450·79-s − 0.109·83-s + 1.27·89-s + 0.419·91-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 7 T + p T^{2} \) | 1.53.h |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 15 T + p T^{2} \) | 1.73.ap |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.56846384678781, −12.76534528548369, −12.43978431188782, −12.19267971282113, −11.52125306681450, −11.19381462333241, −10.48577549205376, −10.15631757254786, −9.582510523738604, −9.162146061290098, −8.769035702815503, −8.064162127436707, −7.844050345466812, −6.848362208125931, −6.603298704373767, −6.379370502153876, −5.679106132277466, −4.926886456202386, −4.431754884037323, −3.975582193588934, −3.574193093595963, −2.657119622539284, −2.078563326386052, −1.772280847426272, −0.6674902293729969, 0,
0.6674902293729969, 1.772280847426272, 2.078563326386052, 2.657119622539284, 3.574193093595963, 3.975582193588934, 4.431754884037323, 4.926886456202386, 5.679106132277466, 6.379370502153876, 6.603298704373767, 6.848362208125931, 7.844050345466812, 8.064162127436707, 8.769035702815503, 9.162146061290098, 9.582510523738604, 10.15631757254786, 10.48577549205376, 11.19381462333241, 11.52125306681450, 12.19267971282113, 12.43978431188782, 12.76534528548369, 13.56846384678781
