| L(s) = 1 | + 2.14·2-s + 21.7·3-s − 27.4·4-s + 25·5-s + 46.5·6-s + 43.1·7-s − 127.·8-s + 229.·9-s + 53.5·10-s + 121·11-s − 595.·12-s − 542.·13-s + 92.4·14-s + 543.·15-s + 604.·16-s + 254.·17-s + 491.·18-s − 361·19-s − 685.·20-s + 937.·21-s + 259.·22-s − 152.·23-s − 2.76e3·24-s + 625·25-s − 1.16e3·26-s − 290.·27-s − 1.18e3·28-s + ⋯ |
| L(s) = 1 | + 0.378·2-s + 1.39·3-s − 0.856·4-s + 0.447·5-s + 0.528·6-s + 0.332·7-s − 0.703·8-s + 0.944·9-s + 0.169·10-s + 0.301·11-s − 1.19·12-s − 0.889·13-s + 0.126·14-s + 0.623·15-s + 0.590·16-s + 0.213·17-s + 0.357·18-s − 0.229·19-s − 0.383·20-s + 0.464·21-s + 0.114·22-s − 0.0602·23-s − 0.980·24-s + 0.200·25-s − 0.336·26-s − 0.0767·27-s − 0.284·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 - 2.14T + 32T^{2} \) |
| 3 | \( 1 - 21.7T + 243T^{2} \) |
| 7 | \( 1 - 43.1T + 1.68e4T^{2} \) |
| 13 | \( 1 + 542.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 254.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 152.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.02e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.42e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.10e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.38e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.81e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.03e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.08e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.42e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.82e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.81e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 9.19e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.60e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.03e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.59e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.16e4T + 8.58e9T^{2} \) |
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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
−8.970797717068035758423517658524, −8.008444218900222666192247874246, −7.41210508048661452131888672813, −6.08416468546978398951158788329, −5.14682988749665244804262873468, −4.25026249942891797503274810343, −3.42161051249426745406769495239, −2.51256742656546150155295824014, −1.51823526501763084947606756442, 0,
1.51823526501763084947606756442, 2.51256742656546150155295824014, 3.42161051249426745406769495239, 4.25026249942891797503274810343, 5.14682988749665244804262873468, 6.08416468546978398951158788329, 7.41210508048661452131888672813, 8.008444218900222666192247874246, 8.970797717068035758423517658524
