| L(s) = 1 | − 2·2-s + 4·4-s − 19.8·5-s − 5.87·7-s − 8·8-s + 39.7·10-s + 18.7·11-s + 45.8·13-s + 11.7·14-s + 16·16-s + 16.8·17-s − 10.3·19-s − 79.4·20-s − 37.4·22-s + 49.8·23-s + 269.·25-s − 91.7·26-s − 23.4·28-s + 10.9·29-s + 151.·31-s − 32·32-s − 33.7·34-s + 116.·35-s + 346.·37-s + 20.7·38-s + 158.·40-s + 264.·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.77·5-s − 0.316·7-s − 0.353·8-s + 1.25·10-s + 0.513·11-s + 0.978·13-s + 0.224·14-s + 0.250·16-s + 0.240·17-s − 0.124·19-s − 0.888·20-s − 0.363·22-s + 0.452·23-s + 2.15·25-s − 0.691·26-s − 0.158·28-s + 0.0698·29-s + 0.878·31-s − 0.176·32-s − 0.170·34-s + 0.563·35-s + 1.53·37-s + 0.0883·38-s + 0.628·40-s + 1.00·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8153125151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8153125151\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 19.8T + 125T^{2} \) |
| 7 | \( 1 + 5.87T + 343T^{2} \) |
| 11 | \( 1 - 18.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 45.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 16.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 10.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 49.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 10.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 151.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 346.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 264.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 411.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 472.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 290.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 53.2T + 2.05e5T^{2} \) |
| 61 | \( 1 + 293.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 398.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 647.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 478.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 374.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 933.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 368.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 274.T + 9.12e5T^{2} \) |
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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
−12.03571401144105409218043838421, −11.42916099519757742877233420313, −10.51743045390285681073524494457, −9.124138357204641761195068770497, −8.254859267019606581004661126327, −7.39637503049579507498152148834, −6.27923160804053056309974071552, −4.32853905527482945936387563174, −3.21010623684492776275209463611, −0.812327752269083132670016101859,
0.812327752269083132670016101859, 3.21010623684492776275209463611, 4.32853905527482945936387563174, 6.27923160804053056309974071552, 7.39637503049579507498152148834, 8.254859267019606581004661126327, 9.124138357204641761195068770497, 10.51743045390285681073524494457, 11.42916099519757742877233420313, 12.03571401144105409218043838421
