| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s − 11-s + 13-s + 14-s + 16-s + 6·17-s + 2·19-s + 20-s − 22-s + 3·23-s + 25-s + 26-s + 28-s − 6·29-s − 4·31-s + 32-s + 6·34-s + 35-s + 2·37-s + 2·38-s + 40-s + 12·41-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 + 0.277·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.458·19-s + 0.223·20-s − 0.213·22-s + 0.625·23-s + 1/5·25-s + 0.196·26-s + 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 1.02·34-s + 0.169·35-s + 0.328·37-s + 0.324·38-s + 0.158·40-s + 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.039307788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.039307788\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−14.05070775797982, −13.28227173578952, −12.94695953405277, −12.48984028427948, −12.01999857327259, −11.35053752820433, −10.98022425585527, −10.57656389796067, −9.914758348617220, −9.283014310038328, −9.130268860176586, −8.027399747053673, −7.810478125888407, −7.332110285588637, −6.621709999653371, −6.045509296177963, −5.437256085285124, −5.306315169815122, −4.536420794961803, −3.849851200799989, −3.369127450471352, −2.713836532995871, −2.091181534623669, −1.353034976890365, −0.7300235034511670,
0.7300235034511670, 1.353034976890365, 2.091181534623669, 2.713836532995871, 3.369127450471352, 3.849851200799989, 4.536420794961803, 5.306315169815122, 5.437256085285124, 6.045509296177963, 6.621709999653371, 7.332110285588637, 7.810478125888407, 8.027399747053673, 9.130268860176586, 9.283014310038328, 9.914758348617220, 10.57656389796067, 10.98022425585527, 11.35053752820433, 12.01999857327259, 12.48984028427948, 12.94695953405277, 13.28227173578952, 14.05070775797982
