L(s) = 1 | − 3·11-s − 4·13-s + 3·17-s − 4·19-s − 2·23-s − 2·29-s − 4·31-s + 4·37-s + 10·41-s + 7·43-s + 4·47-s − 7·49-s + 8·53-s + 59-s − 4·61-s − 12·67-s + 14·71-s + 2·73-s − 83-s − 89-s − 17·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 0.904·11-s − 1.10·13-s + 0.727·17-s − 0.917·19-s − 0.417·23-s − 0.371·29-s − 0.718·31-s + 0.657·37-s + 1.56·41-s + 1.06·43-s + 0.583·47-s − 49-s + 1.09·53-s + 0.130·59-s − 0.512·61-s − 1.46·67-s + 1.66·71-s + 0.234·73-s − 0.109·83-s − 0.105·89-s − 1.72·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−13.75957664901538, −13.15603144700429, −12.65229960812869, −12.43557403101920, −11.93167002395053, −11.17656173413829, −10.86374313728029, −10.37038480165502, −9.839019774423798, −9.412458914434987, −8.945320865919796, −8.194139653309643, −7.803202982675730, −7.433522484089193, −6.894812268579483, −6.178198966357788, −5.639314173844314, −5.319507630056483, −4.539176335915652, −4.178745350844117, −3.482440497870224, −2.630697272846672, −2.444947077807837, −1.656575197466475, −0.7258872010834401, 0,
0.7258872010834401, 1.656575197466475, 2.444947077807837, 2.630697272846672, 3.482440497870224, 4.178745350844117, 4.539176335915652, 5.319507630056483, 5.639314173844314, 6.178198966357788, 6.894812268579483, 7.433522484089193, 7.803202982675730, 8.194139653309643, 8.945320865919796, 9.412458914434987, 9.839019774423798, 10.37038480165502, 10.86374313728029, 11.17656173413829, 11.93167002395053, 12.43557403101920, 12.65229960812869, 13.15603144700429, 13.75957664901538