| L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s − 11-s + 14-s + 16-s − 5·19-s − 20-s − 22-s + 6·23-s + 25-s + 28-s + 7·31-s + 32-s − 35-s − 8·37-s − 5·38-s − 40-s + 6·41-s − 7·43-s − 44-s + 6·46-s − 3·47-s − 6·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.301·11-s + 0.267·14-s + 1/4·16-s − 1.14·19-s − 0.223·20-s − 0.213·22-s + 1.25·23-s + 1/5·25-s + 0.188·28-s + 1.25·31-s + 0.176·32-s − 0.169·35-s − 1.31·37-s − 0.811·38-s − 0.158·40-s + 0.937·41-s − 1.06·43-s − 0.150·44-s + 0.884·46-s − 0.437·47-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 167310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.871181563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.871181563\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 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
−13.14799612234204, −12.85086027088679, −12.33547614621681, −11.79660741892799, −11.49032402895973, −10.84967176454875, −10.58481908846448, −10.10111314340325, −9.351824839008896, −8.878062659548785, −8.214686712118782, −8.053074589200280, −7.314937872979878, −6.819093407156542, −6.444500012937060, −5.837663928694660, −5.136864423494796, −4.823623597568468, −4.313739316691132, −3.771055145334060, −3.023500276833585, −2.755198820560159, −1.874089754087199, −1.361779424667992, −0.4256606693468600,
0.4256606693468600, 1.361779424667992, 1.874089754087199, 2.755198820560159, 3.023500276833585, 3.771055145334060, 4.313739316691132, 4.823623597568468, 5.136864423494796, 5.837663928694660, 6.444500012937060, 6.819093407156542, 7.314937872979878, 8.053074589200280, 8.214686712118782, 8.878062659548785, 9.351824839008896, 10.10111314340325, 10.58481908846448, 10.84967176454875, 11.49032402895973, 11.79660741892799, 12.33547614621681, 12.85086027088679, 13.14799612234204
