| L(s) = 1 | − 9.30·2-s − 41.3·4-s − 125·5-s + 343·7-s + 1.57e3·8-s + 1.16e3·10-s + 6.78e3·11-s + 1.40e4·13-s − 3.19e3·14-s − 9.37e3·16-s + 1.77e4·17-s + 4.77e4·19-s + 5.17e3·20-s − 6.31e4·22-s + 2.46e3·23-s + 1.56e4·25-s − 1.30e5·26-s − 1.41e4·28-s − 7.28e4·29-s + 2.60e5·31-s − 1.14e5·32-s − 1.65e5·34-s − 4.28e4·35-s + 1.18e5·37-s − 4.44e5·38-s − 1.97e5·40-s − 4.34e5·41-s + ⋯ |
| L(s) = 1 | − 0.822·2-s − 0.323·4-s − 0.447·5-s + 0.377·7-s + 1.08·8-s + 0.367·10-s + 1.53·11-s + 1.77·13-s − 0.310·14-s − 0.572·16-s + 0.875·17-s + 1.59·19-s + 0.144·20-s − 1.26·22-s + 0.0423·23-s + 0.199·25-s − 1.45·26-s − 0.122·28-s − 0.554·29-s + 1.56·31-s − 0.617·32-s − 0.720·34-s − 0.169·35-s + 0.384·37-s − 1.31·38-s − 0.486·40-s − 0.984·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.794395175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.794395175\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 125T \) |
| 7 | \( 1 - 343T \) |
| good | 2 | \( 1 + 9.30T + 128T^{2} \) |
| 11 | \( 1 - 6.78e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.40e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.77e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.77e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.46e3T + 3.40e9T^{2} \) |
| 29 | \( 1 + 7.28e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.60e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.18e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.34e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.99e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.47e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.40e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.96e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 7.26e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 7.56e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.50e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 6.38e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.43e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.38e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 7.59e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.91e6T + 8.07e13T^{2} \) |
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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
−10.26696128029340799595192387922, −9.379349229783479918221291672611, −8.598403959248894104954904199787, −7.88532942780285555606018948068, −6.81260410753512156555785062081, −5.56967829751074843299601672745, −4.24114520651057057163951181784, −3.44644213431197640017499844243, −1.32582429247436369630130806086, −0.926224303801033086956288893245,
0.926224303801033086956288893245, 1.32582429247436369630130806086, 3.44644213431197640017499844243, 4.24114520651057057163951181784, 5.56967829751074843299601672745, 6.81260410753512156555785062081, 7.88532942780285555606018948068, 8.598403959248894104954904199787, 9.379349229783479918221291672611, 10.26696128029340799595192387922
