| L(s) = 1 | + 4·5-s − 7-s − 2·11-s − 2·13-s + 4·19-s − 6·23-s + 11·25-s + 10·29-s + 8·31-s − 4·35-s + 10·37-s + 4·41-s + 8·43-s − 4·47-s + 49-s − 10·53-s − 8·55-s + 8·59-s − 6·61-s − 8·65-s − 4·67-s + 14·71-s + 6·73-s + 2·77-s − 4·79-s − 12·83-s − 4·89-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.377·7-s − 0.603·11-s − 0.554·13-s + 0.917·19-s − 1.25·23-s + 11/5·25-s + 1.85·29-s + 1.43·31-s − 0.676·35-s + 1.64·37-s + 0.624·41-s + 1.21·43-s − 0.583·47-s + 1/7·49-s − 1.37·53-s − 1.07·55-s + 1.04·59-s − 0.768·61-s − 0.992·65-s − 0.488·67-s + 1.66·71-s + 0.702·73-s + 0.227·77-s − 0.450·79-s − 1.31·83-s − 0.423·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.354652703\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.354652703\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−9.481276071511481565581284791220, −8.409578506072020402089900906100, −7.61971828815925419087584402587, −6.49198050199370736249656456104, −6.07899407066809115934805106887, −5.24330200469260993508502492214, −4.44802041941911311166014127039, −2.86972543343592766466945454261, −2.39751668918426191936188305346, −1.07233750136519077963315820080,
1.07233750136519077963315820080, 2.39751668918426191936188305346, 2.86972543343592766466945454261, 4.44802041941911311166014127039, 5.24330200469260993508502492214, 6.07899407066809115934805106887, 6.49198050199370736249656456104, 7.61971828815925419087584402587, 8.409578506072020402089900906100, 9.481276071511481565581284791220
