| L(s) = 1 | − 0.232·2-s − 7.94·4-s − 7.75·5-s − 7·7-s + 3.71·8-s + 1.80·10-s − 11·11-s + 59.4·13-s + 1.63·14-s + 62.7·16-s + 45.4·17-s + 111.·19-s + 61.6·20-s + 2.56·22-s − 105.·23-s − 64.8·25-s − 13.8·26-s + 55.6·28-s − 10.0·29-s + 315.·31-s − 44.3·32-s − 10.5·34-s + 54.3·35-s − 182.·37-s − 26.0·38-s − 28.8·40-s − 487.·41-s + ⋯ |
| L(s) = 1 | − 0.0823·2-s − 0.993·4-s − 0.693·5-s − 0.377·7-s + 0.164·8-s + 0.0571·10-s − 0.301·11-s + 1.26·13-s + 0.0311·14-s + 0.979·16-s + 0.648·17-s + 1.34·19-s + 0.689·20-s + 0.0248·22-s − 0.954·23-s − 0.518·25-s − 0.104·26-s + 0.375·28-s − 0.0641·29-s + 1.82·31-s − 0.244·32-s − 0.0533·34-s + 0.262·35-s − 0.809·37-s − 0.111·38-s − 0.113·40-s − 1.85·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 0.232T + 8T^{2} \) |
| 5 | \( 1 + 7.75T + 125T^{2} \) |
| 13 | \( 1 - 59.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 45.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 111.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 105.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 10.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 315.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 182.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 487.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 358.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 205.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 134.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 891.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 654.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 102.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 119.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 346.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 774.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.04e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 502.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 939.T + 9.12e5T^{2} \) |
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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
−9.929762048472005608922200479549, −8.416569075101449374280467584325, −8.301127173889242612819321348991, −7.10805718346756694460687423744, −5.91543818001065545160439451410, −5.03224414784603519334094480084, −3.87313705832006731656763319464, −3.26307181189100111763425618873, −1.24274805081644487000624017387, 0,
1.24274805081644487000624017387, 3.26307181189100111763425618873, 3.87313705832006731656763319464, 5.03224414784603519334094480084, 5.91543818001065545160439451410, 7.10805718346756694460687423744, 8.301127173889242612819321348991, 8.416569075101449374280467584325, 9.929762048472005608922200479549
