| L(s) = 1 | − 4·7-s + 6·11-s + 4·13-s − 3·17-s + 7·19-s + 9·23-s + 7·31-s − 2·37-s − 6·41-s + 2·43-s + 9·49-s + 9·53-s − 12·59-s − 7·61-s + 2·67-s + 6·71-s − 2·73-s − 24·77-s + 79-s − 9·83-s + 6·89-s − 16·91-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 1.80·11-s + 1.10·13-s − 0.727·17-s + 1.60·19-s + 1.87·23-s + 1.25·31-s − 0.328·37-s − 0.937·41-s + 0.304·43-s + 9/7·49-s + 1.23·53-s − 1.56·59-s − 0.896·61-s + 0.244·67-s + 0.712·71-s − 0.234·73-s − 2.73·77-s + 0.112·79-s − 0.987·83-s + 0.635·89-s − 1.67·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.327865276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.327865276\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−16.62680347662897, −15.90695253684528, −15.47479735174780, −14.98257610766646, −13.99374616772559, −13.69643306058531, −13.20973801330491, −12.46861478402799, −11.92267839621329, −11.37713582480028, −10.76564578145262, −9.947747959284723, −9.379165502155909, −9.014282567655022, −8.470906990903180, −7.317814906424539, −6.789005133119176, −6.401966853674812, −5.762583830952507, −4.813324027085895, −3.965216657212247, −3.342831635639116, −2.876348027843199, −1.449701929719029, −0.7876065460097878,
0.7876065460097878, 1.449701929719029, 2.876348027843199, 3.342831635639116, 3.965216657212247, 4.813324027085895, 5.762583830952507, 6.401966853674812, 6.789005133119176, 7.317814906424539, 8.470906990903180, 9.014282567655022, 9.379165502155909, 9.947747959284723, 10.76564578145262, 11.37713582480028, 11.92267839621329, 12.46861478402799, 13.20973801330491, 13.69643306058531, 13.99374616772559, 14.98257610766646, 15.47479735174780, 15.90695253684528, 16.62680347662897
