| L(s) = 1 | − 0.561·2-s − 7.68·4-s − 18.6·5-s + 7·7-s + 8.80·8-s + 10.4·10-s + 11·11-s + 36.4·13-s − 3.93·14-s + 56.5·16-s − 41.1·17-s − 23.6·19-s + 143.·20-s − 6.17·22-s + 140.·23-s + 224.·25-s − 20.4·26-s − 53.7·28-s − 278.·29-s + 191.·31-s − 102.·32-s + 23.0·34-s − 130.·35-s + 196.·37-s + 13.3·38-s − 164.·40-s + 322.·41-s + ⋯ |
| L(s) = 1 | − 0.198·2-s − 0.960·4-s − 1.67·5-s + 0.377·7-s + 0.389·8-s + 0.331·10-s + 0.301·11-s + 0.777·13-s − 0.0750·14-s + 0.883·16-s − 0.586·17-s − 0.286·19-s + 1.60·20-s − 0.0598·22-s + 1.26·23-s + 1.79·25-s − 0.154·26-s − 0.363·28-s − 1.78·29-s + 1.10·31-s − 0.564·32-s + 0.116·34-s − 0.631·35-s + 0.871·37-s + 0.0568·38-s − 0.650·40-s + 1.22·41-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 | \( 1 + 0.561T + 8T^{2} \) |
| 5 | \( 1 + 18.6T + 125T^{2} \) |
| 13 | \( 1 - 36.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 41.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 23.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 140.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 278.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 191.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 196.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 322.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 3.67T + 7.95e4T^{2} \) |
| 47 | \( 1 - 397.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 597.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 668.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 667.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 730.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 31.2T + 3.57e5T^{2} \) |
| 73 | \( 1 + 434.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 782.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 426.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 899.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 942.T + 9.12e5T^{2} \) |
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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
−9.291499249620192903756288632822, −8.800852156284164029161057675573, −7.911556582000346082722831442695, −7.37463526735258782032998359880, −6.02916609716696278955398001203, −4.64842969665414758982860636284, −4.19144808374299801080415889668, −3.19288720211848933869566176116, −1.14519559924680600583646109908, 0,
1.14519559924680600583646109908, 3.19288720211848933869566176116, 4.19144808374299801080415889668, 4.64842969665414758982860636284, 6.02916609716696278955398001203, 7.37463526735258782032998359880, 7.911556582000346082722831442695, 8.800852156284164029161057675573, 9.291499249620192903756288632822
