| L(s) = 1 | − 0.633·2-s + 19.5·3-s − 31.5·4-s + 25·5-s − 12.3·6-s + 40.2·8-s + 137.·9-s − 15.8·10-s + 76.2·11-s − 616.·12-s + 266.·13-s + 487.·15-s + 985.·16-s + 267.·17-s − 87.3·18-s − 1.76e3·19-s − 789.·20-s − 48.2·22-s + 4.41e3·23-s + 786.·24-s + 625·25-s − 168.·26-s − 2.05e3·27-s + 129.·29-s − 309.·30-s + 5.98e3·31-s − 1.91e3·32-s + ⋯ |
| L(s) = 1 | − 0.111·2-s + 1.25·3-s − 0.987·4-s + 0.447·5-s − 0.140·6-s + 0.222·8-s + 0.567·9-s − 0.0500·10-s + 0.189·11-s − 1.23·12-s + 0.437·13-s + 0.559·15-s + 0.962·16-s + 0.224·17-s − 0.0635·18-s − 1.12·19-s − 0.441·20-s − 0.0212·22-s + 1.73·23-s + 0.278·24-s + 0.200·25-s − 0.0489·26-s − 0.541·27-s + 0.0284·29-s − 0.0626·30-s + 1.11·31-s − 0.330·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.682867576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.682867576\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 0.633T + 32T^{2} \) |
| 3 | \( 1 - 19.5T + 243T^{2} \) |
| 11 | \( 1 - 76.2T + 1.61e5T^{2} \) |
| 13 | \( 1 - 266.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 267.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.76e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.41e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 129.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.98e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.90e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.26e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.78e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.51e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.06e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.36e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.22e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.19e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.95e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.41e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.07e5T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.51e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.97e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.31e5T + 8.58e9T^{2} \) |
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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
−11.01255479227250322896114680440, −9.944128748960689494622675497599, −9.063140046798552895242753998554, −8.601083920234288462725862401642, −7.60796546614083431980048567717, −6.16340076812794854373663611831, −4.78798496633865115228398757474, −3.68242920103125580006973961807, −2.51836285509172468411209319946, −0.979618934956148576040552675077,
0.979618934956148576040552675077, 2.51836285509172468411209319946, 3.68242920103125580006973961807, 4.78798496633865115228398757474, 6.16340076812794854373663611831, 7.60796546614083431980048567717, 8.601083920234288462725862401642, 9.063140046798552895242753998554, 9.944128748960689494622675497599, 11.01255479227250322896114680440
