| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s + 11-s + 13-s − 14-s + 16-s + 4·17-s − 20-s − 22-s − 8·23-s + 25-s − 26-s + 28-s − 6·29-s − 32-s − 4·34-s − 35-s − 6·37-s + 40-s + 10·41-s − 4·43-s + 44-s + 8·46-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 + 0.970·17-s − 0.223·20-s − 0.213·22-s − 1.66·23-s + 1/5·25-s − 0.196·26-s + 0.188·28-s − 1.11·29-s − 0.176·32-s − 0.685·34-s − 0.169·35-s − 0.986·37-s + 0.158·40-s + 1.56·41-s − 0.609·43-s + 0.150·44-s + 1.17·46-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
| \(L(1)\) |
\(\approx\) |
\(0.9455279163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9455279163\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−13.96797846253305, −13.46514310696083, −12.62670532974271, −12.29717240778893, −11.90088741763909, −11.24776064600706, −10.99695013752453, −10.35456469084786, −9.832771465262029, −9.416697425337577, −8.862601917175634, −8.114347915259417, −8.031160611548730, −7.426006502461160, −6.865144859534437, −6.253971768430559, −5.658292049409645, −5.257196150407468, −4.296104943918565, −3.913999129697212, −3.285600023824599, −2.580576883752309, −1.742500089572896, −1.346623739917198, −0.3549220529284713,
0.3549220529284713, 1.346623739917198, 1.742500089572896, 2.580576883752309, 3.285600023824599, 3.913999129697212, 4.296104943918565, 5.257196150407468, 5.658292049409645, 6.253971768430559, 6.865144859534437, 7.426006502461160, 8.031160611548730, 8.114347915259417, 8.862601917175634, 9.416697425337577, 9.832771465262029, 10.35456469084786, 10.99695013752453, 11.24776064600706, 11.90088741763909, 12.29717240778893, 12.62670532974271, 13.46514310696083, 13.96797846253305
