| L(s) = 1 | − 0.183·2-s − 7.96·4-s + 17.9·5-s − 7·7-s + 2.92·8-s − 3.28·10-s + 11·11-s − 53.2·13-s + 1.28·14-s + 63.1·16-s + 86.7·17-s + 55.6·19-s − 142.·20-s − 2.01·22-s − 185.·23-s + 196.·25-s + 9.75·26-s + 55.7·28-s + 145.·29-s + 132.·31-s − 35.0·32-s − 15.9·34-s − 125.·35-s − 129.·37-s − 10.2·38-s + 52.4·40-s − 139.·41-s + ⋯ |
| L(s) = 1 | − 0.0648·2-s − 0.995·4-s + 1.60·5-s − 0.377·7-s + 0.129·8-s − 0.103·10-s + 0.301·11-s − 1.13·13-s + 0.0245·14-s + 0.987·16-s + 1.23·17-s + 0.672·19-s − 1.59·20-s − 0.0195·22-s − 1.67·23-s + 1.57·25-s + 0.0736·26-s + 0.376·28-s + 0.934·29-s + 0.768·31-s − 0.193·32-s − 0.0802·34-s − 0.606·35-s − 0.574·37-s − 0.0435·38-s + 0.207·40-s − 0.532·41-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
| \(L(2)\) |
\(\approx\) |
\(1.966097713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.966097713\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 0.183T + 8T^{2} \) |
| 5 | \( 1 - 17.9T + 125T^{2} \) |
| 13 | \( 1 + 53.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 86.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 55.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 185.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 145.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 132.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 129.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 139.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 87.1T + 7.95e4T^{2} \) |
| 47 | \( 1 - 576.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 92.0T + 1.48e5T^{2} \) |
| 59 | \( 1 + 315.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 146.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 734.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 702.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 361.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.30e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.38e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 212.T + 9.12e5T^{2} \) |
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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
−9.894647240996706169396942381018, −9.522439691522870505063731410454, −8.519780591081875397332684768419, −7.50586551991173587176439995998, −6.28265185878199681384401012433, −5.54511974197424530591092054552, −4.76699656143625685901496365558, −3.42680860901895258361254926813, −2.16833422925059465524448182676, −0.843632661861296422724326384891,
0.843632661861296422724326384891, 2.16833422925059465524448182676, 3.42680860901895258361254926813, 4.76699656143625685901496365558, 5.54511974197424530591092054552, 6.28265185878199681384401012433, 7.50586551991173587176439995998, 8.519780591081875397332684768419, 9.522439691522870505063731410454, 9.894647240996706169396942381018
