| L(s) = 1 | − 2-s + 4-s + 5-s − 4·7-s − 8-s − 10-s − 4·11-s + 13-s + 4·14-s + 16-s + 8·19-s + 20-s + 4·22-s − 4·23-s + 25-s − 26-s − 4·28-s + 2·29-s − 32-s − 4·35-s − 6·37-s − 8·38-s − 40-s − 6·41-s + 4·43-s − 4·44-s + 4·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s − 0.353·8-s − 0.316·10-s − 1.20·11-s + 0.277·13-s + 1.06·14-s + 1/4·16-s + 1.83·19-s + 0.223·20-s + 0.852·22-s − 0.834·23-s + 1/5·25-s − 0.196·26-s − 0.755·28-s + 0.371·29-s − 0.176·32-s − 0.676·35-s − 0.986·37-s − 1.29·38-s − 0.158·40-s − 0.937·41-s + 0.609·43-s − 0.603·44-s + 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6429841242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6429841242\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−12.63168277317577, −12.10618208069250, −11.67996555079575, −11.14546296471810, −10.53390805543971, −10.09137752961797, −9.979857222080991, −9.484096808093842, −9.008322613376592, −8.516677554144256, −7.998841583951163, −7.495943771228038, −7.029652576178900, −6.667138231568855, −6.020740408632501, −5.653853965452211, −5.260294341721792, −4.606207391150622, −3.653476515127619, −3.427015909796814, −2.758130099723928, −2.470451117858300, −1.652353819633868, −1.003133740096123, −0.2618221328788463,
0.2618221328788463, 1.003133740096123, 1.652353819633868, 2.470451117858300, 2.758130099723928, 3.427015909796814, 3.653476515127619, 4.606207391150622, 5.260294341721792, 5.653853965452211, 6.020740408632501, 6.667138231568855, 7.029652576178900, 7.495943771228038, 7.998841583951163, 8.516677554144256, 9.008322613376592, 9.484096808093842, 9.979857222080991, 10.09137752961797, 10.53390805543971, 11.14546296471810, 11.67996555079575, 12.10618208069250, 12.63168277317577
