L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s − 11-s + 13-s + 16-s − 2·17-s + 2·19-s + 20-s + 22-s + 2·23-s − 4·25-s − 26-s + 7·29-s − 3·31-s − 32-s + 2·34-s − 2·37-s − 2·38-s − 40-s + 8·41-s − 8·43-s − 44-s − 2·46-s + 4·50-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 0.301·11-s + 0.277·13-s + 1/4·16-s − 0.485·17-s + 0.458·19-s + 0.223·20-s + 0.213·22-s + 0.417·23-s − 4/5·25-s − 0.196·26-s + 1.29·29-s − 0.538·31-s − 0.176·32-s + 0.342·34-s − 0.328·37-s − 0.324·38-s − 0.158·40-s + 1.24·41-s − 1.21·43-s − 0.150·44-s − 0.294·46-s + 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.485255647\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.485255647\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 - 11 T + p T^{2} \) | 1.83.al |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−16.46788713588524, −15.89974658587799, −15.49963672640828, −14.83940352921363, −14.11515154722356, −13.63501996612062, −13.02732313961599, −12.37718134197768, −11.72874624063226, −11.15214630758491, −10.54950188811940, −10.03204483970442, −9.382865240541641, −8.914226837515496, −8.165374598319363, −7.688820691285508, −6.846300894267944, −6.385111037881819, −5.598364691625594, −4.982955985141765, −4.048677799601795, −3.173775758291352, −2.403076986657317, −1.624190808141456, −0.6319272388518665,
0.6319272388518665, 1.624190808141456, 2.403076986657317, 3.173775758291352, 4.048677799601795, 4.982955985141765, 5.598364691625594, 6.385111037881819, 6.846300894267944, 7.688820691285508, 8.165374598319363, 8.914226837515496, 9.382865240541641, 10.03204483970442, 10.54950188811940, 11.15214630758491, 11.72874624063226, 12.37718134197768, 13.02732313961599, 13.63501996612062, 14.11515154722356, 14.83940352921363, 15.49963672640828, 15.89974658587799, 16.46788713588524