| L(s) = 1 | + 3-s − 2·4-s − 5-s + 2·7-s + 9-s + 5·11-s − 2·12-s − 15-s + 4·16-s + 5·17-s + 6·19-s + 2·20-s + 2·21-s − 9·23-s + 25-s + 27-s − 4·28-s + 8·29-s + 5·31-s + 5·33-s − 2·35-s − 2·36-s − 8·37-s + 7·41-s − 43-s − 10·44-s − 45-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.50·11-s − 0.577·12-s − 0.258·15-s + 16-s + 1.21·17-s + 1.37·19-s + 0.447·20-s + 0.436·21-s − 1.87·23-s + 1/5·25-s + 0.192·27-s − 0.755·28-s + 1.48·29-s + 0.898·31-s + 0.870·33-s − 0.338·35-s − 1/3·36-s − 1.31·37-s + 1.09·41-s − 0.152·43-s − 1.50·44-s − 0.149·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 109005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.606723761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.606723761\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| 43 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 9 T + p T^{2} \) | 1.67.j |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 3 T + p T^{2} \) | 1.97.d |
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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
−13.68804459468213, −13.58888113792933, −12.43874686706203, −12.18664362647693, −11.92665995629581, −11.44965345216368, −10.51056955784608, −10.13976414999938, −9.720559616309617, −9.096338350055355, −8.803645138526821, −8.169868003230988, −7.755371698439830, −7.519607286253587, −6.585313787193369, −6.100439864384500, −5.412075895609853, −4.862200648257398, −4.309704941363869, −3.868277248859232, −3.397470169211885, −2.785074046841905, −1.734963589720550, −1.205298725133032, −0.6654712915163429,
0.6654712915163429, 1.205298725133032, 1.734963589720550, 2.785074046841905, 3.397470169211885, 3.868277248859232, 4.309704941363869, 4.862200648257398, 5.412075895609853, 6.100439864384500, 6.585313787193369, 7.519607286253587, 7.755371698439830, 8.169868003230988, 8.803645138526821, 9.096338350055355, 9.720559616309617, 10.13976414999938, 10.51056955784608, 11.44965345216368, 11.92665995629581, 12.18664362647693, 12.43874686706203, 13.58888113792933, 13.68804459468213
