| L(s) = 1 | + 2·5-s − 4·7-s − 2·13-s − 6·17-s + 4·23-s − 25-s + 29-s + 6·31-s − 8·35-s − 4·37-s − 6·41-s − 4·43-s + 9·49-s − 2·53-s + 2·59-s − 4·65-s − 12·67-s + 14·73-s + 10·79-s − 6·83-s − 12·85-s + 6·89-s + 8·91-s + 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s − 0.554·13-s − 1.45·17-s + 0.834·23-s − 1/5·25-s + 0.185·29-s + 1.07·31-s − 1.35·35-s − 0.657·37-s − 0.937·41-s − 0.609·43-s + 9/7·49-s − 0.274·53-s + 0.260·59-s − 0.496·65-s − 1.46·67-s + 1.63·73-s + 1.12·79-s − 0.658·83-s − 1.30·85-s + 0.635·89-s + 0.838·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.285515845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.285515845\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 29 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−15.86615881167462, −15.35675504037542, −14.99704217531325, −13.95120396653420, −13.72015799001638, −13.10363081254330, −12.80467204644663, −12.06220785693683, −11.50945842150411, −10.65126066942505, −10.18765820361773, −9.720495886487554, −9.138115747157386, −8.750765693660950, −7.850575450770791, −6.968603752429377, −6.552804277657389, −6.202701442949699, −5.305567273268509, −4.750305648867072, −3.859369999132551, −3.069776696956202, −2.498946007865051, −1.723119859750669, −0.4616014127767946,
0.4616014127767946, 1.723119859750669, 2.498946007865051, 3.069776696956202, 3.859369999132551, 4.750305648867072, 5.305567273268509, 6.202701442949699, 6.552804277657389, 6.968603752429377, 7.850575450770791, 8.750765693660950, 9.138115747157386, 9.720495886487554, 10.18765820361773, 10.65126066942505, 11.50945842150411, 12.06220785693683, 12.80467204644663, 13.10363081254330, 13.72015799001638, 13.95120396653420, 14.99704217531325, 15.35675504037542, 15.86615881167462
