| L(s) = 1 | − 3·3-s + 2·5-s + 6·9-s + 5·11-s − 7·13-s − 6·15-s − 17-s + 2·19-s − 25-s − 9·27-s + 6·29-s − 4·31-s − 15·33-s − 8·37-s + 21·39-s + 2·41-s + 8·43-s + 12·45-s + 10·47-s + 3·51-s − 3·53-s + 10·55-s − 6·57-s + 12·61-s − 14·65-s + 2·67-s + 71-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.894·5-s + 2·9-s + 1.50·11-s − 1.94·13-s − 1.54·15-s − 0.242·17-s + 0.458·19-s − 1/5·25-s − 1.73·27-s + 1.11·29-s − 0.718·31-s − 2.61·33-s − 1.31·37-s + 3.36·39-s + 0.312·41-s + 1.21·43-s + 1.78·45-s + 1.45·47-s + 0.420·51-s − 0.412·53-s + 1.34·55-s − 0.794·57-s + 1.53·61-s − 1.73·65-s + 0.244·67-s + 0.118·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.167375375\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.167375375\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
| 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
−7.66860733927778170460326291899, −6.90848186518129762938140417434, −6.64342050155731907240401695264, −5.74388561899388298258927307182, −5.34586573669869425869504598290, −4.61127700800814329376249184544, −3.91473285314472524723164861498, −2.50583074099096839702566189393, −1.60506532101432445390514842673, −0.62999088447360887739632380382,
0.62999088447360887739632380382, 1.60506532101432445390514842673, 2.50583074099096839702566189393, 3.91473285314472524723164861498, 4.61127700800814329376249184544, 5.34586573669869425869504598290, 5.74388561899388298258927307182, 6.64342050155731907240401695264, 6.90848186518129762938140417434, 7.66860733927778170460326291899
