| L(s) = 1 | − 2.74·2-s + 4.96·3-s − 0.461·4-s − 5·5-s − 13.6·6-s − 34.3·7-s + 23.2·8-s − 2.37·9-s + 13.7·10-s + 11·11-s − 2.28·12-s − 62.6·13-s + 94.3·14-s − 24.8·15-s − 60.0·16-s − 13.5·17-s + 6.51·18-s + 19·19-s + 2.30·20-s − 170.·21-s − 30.2·22-s − 17.5·23-s + 115.·24-s + 25·25-s + 172.·26-s − 145.·27-s + 15.8·28-s + ⋯ |
| L(s) = 1 | − 0.970·2-s + 0.955·3-s − 0.0576·4-s − 0.447·5-s − 0.927·6-s − 1.85·7-s + 1.02·8-s − 0.0879·9-s + 0.434·10-s + 0.301·11-s − 0.0550·12-s − 1.33·13-s + 1.80·14-s − 0.427·15-s − 0.939·16-s − 0.193·17-s + 0.0853·18-s + 0.229·19-s + 0.0257·20-s − 1.77·21-s − 0.292·22-s − 0.158·23-s + 0.980·24-s + 0.200·25-s + 1.29·26-s − 1.03·27-s + 0.106·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3223780918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3223780918\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 + 2.74T + 8T^{2} \) |
| 3 | \( 1 - 4.96T + 27T^{2} \) |
| 7 | \( 1 + 34.3T + 343T^{2} \) |
| 13 | \( 1 + 62.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 13.5T + 4.91e3T^{2} \) |
| 23 | \( 1 + 17.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 178.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 26.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 308.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 108.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 192.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 335.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 607.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 42.4T + 2.05e5T^{2} \) |
| 61 | \( 1 - 546.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 538.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 253.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.11e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 723.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.02e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 380.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 879.T + 9.12e5T^{2} \) |
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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
−9.421310470708000454465052118056, −8.964328177722430012908382700282, −7.990463155211435977572711428499, −7.32224022208029075785259279520, −6.53969276888985931291479925358, −5.18570302201623675983318408481, −3.87937160932871962005573767058, −3.18048980387619719065622202311, −2.07850574534501653969872803240, −0.31211699734880620858129571012,
0.31211699734880620858129571012, 2.07850574534501653969872803240, 3.18048980387619719065622202311, 3.87937160932871962005573767058, 5.18570302201623675983318408481, 6.53969276888985931291479925358, 7.32224022208029075785259279520, 7.990463155211435977572711428499, 8.964328177722430012908382700282, 9.421310470708000454465052118056
