| L(s) = 1 | − 3.62·2-s + 5.17·4-s + 4.84·5-s + 10.2·8-s − 17.5·10-s − 20.1·11-s + 72.2·13-s − 78.6·16-s + 132.·17-s + 76.9·19-s + 25.0·20-s + 73.0·22-s − 22.4·23-s − 101.·25-s − 262.·26-s + 193.·29-s + 89.7·31-s + 203.·32-s − 480.·34-s + 47.9·37-s − 279.·38-s + 49.6·40-s + 3.41·41-s − 168.·43-s − 104.·44-s + 81.5·46-s + 163.·47-s + ⋯ |
| L(s) = 1 | − 1.28·2-s + 0.646·4-s + 0.433·5-s + 0.453·8-s − 0.555·10-s − 0.551·11-s + 1.54·13-s − 1.22·16-s + 1.88·17-s + 0.928·19-s + 0.280·20-s + 0.708·22-s − 0.203·23-s − 0.812·25-s − 1.97·26-s + 1.23·29-s + 0.520·31-s + 1.12·32-s − 2.42·34-s + 0.212·37-s − 1.19·38-s + 0.196·40-s + 0.0130·41-s − 0.597·43-s − 0.356·44-s + 0.261·46-s + 0.506·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.369241540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.369241540\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 3.62T + 8T^{2} \) |
| 5 | \( 1 - 4.84T + 125T^{2} \) |
| 11 | \( 1 + 20.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 72.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 132.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 76.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 22.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 193.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 89.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 47.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 3.41T + 6.89e4T^{2} \) |
| 43 | \( 1 + 168.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 163.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 337.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 517.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 424.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 978.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 40.4T + 3.57e5T^{2} \) |
| 73 | \( 1 - 482.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.07e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 811.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 997.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.45e3T + 9.12e5T^{2} \) |
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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
−9.290549771325094497698059541275, −8.389277054240070183558111763720, −7.962356323089872201793313561246, −7.08466907486733435823596208125, −6.00685415061052921525557230452, −5.29105513808097321887272781681, −3.97215174945320474629585742124, −2.85947737409268625486610409180, −1.49022794145752444118028556562, −0.801622842332916639903454678476,
0.801622842332916639903454678476, 1.49022794145752444118028556562, 2.85947737409268625486610409180, 3.97215174945320474629585742124, 5.29105513808097321887272781681, 6.00685415061052921525557230452, 7.08466907486733435823596208125, 7.962356323089872201793313561246, 8.389277054240070183558111763720, 9.290549771325094497698059541275
