| L(s) = 1 | − 1.21·2-s + 9·3-s − 30.5·4-s + 0.914·5-s − 10.9·6-s + 61.7·7-s + 76.0·8-s + 81·9-s − 1.11·10-s − 311.·11-s − 274.·12-s − 150.·13-s − 75.1·14-s + 8.23·15-s + 884.·16-s + 1.23e3·17-s − 98.5·18-s − 266.·19-s − 27.9·20-s + 555.·21-s + 378.·22-s − 1.92e3·23-s + 684.·24-s − 3.12e3·25-s + 182.·26-s + 729·27-s − 1.88e3·28-s + ⋯ |
| L(s) = 1 | − 0.215·2-s + 0.577·3-s − 0.953·4-s + 0.0163·5-s − 0.124·6-s + 0.476·7-s + 0.420·8-s + 0.333·9-s − 0.00351·10-s − 0.776·11-s − 0.550·12-s − 0.246·13-s − 0.102·14-s + 0.00944·15-s + 0.863·16-s + 1.03·17-s − 0.0716·18-s − 0.169·19-s − 0.0156·20-s + 0.275·21-s + 0.166·22-s − 0.756·23-s + 0.242·24-s − 0.999·25-s + 0.0530·26-s + 0.192·27-s − 0.454·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 59 | \( 1 - 3.48e3T \) |
| good | 2 | \( 1 + 1.21T + 32T^{2} \) |
| 5 | \( 1 - 0.914T + 3.12e3T^{2} \) |
| 7 | \( 1 - 61.7T + 1.68e4T^{2} \) |
| 11 | \( 1 + 311.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 150.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.23e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 266.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.92e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.98e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.35e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.50e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.39e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.12e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.48e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.53e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 4.68e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.76e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.64e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.62e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.15e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.88e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.51e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.63e4T + 8.58e9T^{2} \) |
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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
−11.18216089591956870585904961600, −9.990519371806501783544803430321, −9.361037089490606555391425866479, −8.066900405628620651623551043060, −7.69058322791114410644009812493, −5.74481078603826255624843191723, −4.63971921250433928994951929363, −3.41499865368432585025667328071, −1.72548427306702597091183689824, 0,
1.72548427306702597091183689824, 3.41499865368432585025667328071, 4.63971921250433928994951929363, 5.74481078603826255624843191723, 7.69058322791114410644009812493, 8.066900405628620651623551043060, 9.361037089490606555391425866479, 9.990519371806501783544803430321, 11.18216089591956870585904961600
