| L(s) = 1 | − 2-s + 3-s − 4-s − 6-s + 5·7-s + 3·8-s + 9-s − 12-s − 4·13-s − 5·14-s − 16-s − 2·17-s − 18-s + 2·19-s + 5·21-s + 23-s + 3·24-s + 4·26-s + 27-s − 5·28-s + 5·29-s + 9·31-s − 5·32-s + 2·34-s − 36-s − 3·37-s − 2·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.88·7-s + 1.06·8-s + 1/3·9-s − 0.288·12-s − 1.10·13-s − 1.33·14-s − 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.458·19-s + 1.09·21-s + 0.208·23-s + 0.612·24-s + 0.784·26-s + 0.192·27-s − 0.944·28-s + 0.928·29-s + 1.61·31-s − 0.883·32-s + 0.342·34-s − 1/6·36-s − 0.493·37-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.551094641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.551094641\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−13.11470843306325, −12.48107579701342, −12.03250031122544, −11.57414980412526, −11.10837144310971, −10.48909684983678, −10.18558428887296, −9.736322532156629, −9.034568174286931, −8.838958262167027, −8.230658744436851, −7.856438801033377, −7.692577811712584, −6.955718527465343, −6.531359759046340, −5.537783773364040, −4.999867727693916, −4.730760552036400, −4.373301594423581, −3.678452121536456, −2.847271648255805, −2.304974045374615, −1.704654549548936, −1.187918172448643, −0.5306510885555749,
0.5306510885555749, 1.187918172448643, 1.704654549548936, 2.304974045374615, 2.847271648255805, 3.678452121536456, 4.373301594423581, 4.730760552036400, 4.999867727693916, 5.537783773364040, 6.531359759046340, 6.955718527465343, 7.692577811712584, 7.856438801033377, 8.230658744436851, 8.838958262167027, 9.034568174286931, 9.736322532156629, 10.18558428887296, 10.48909684983678, 11.10837144310971, 11.57414980412526, 12.03250031122544, 12.48107579701342, 13.11470843306325
