| L(s) = 1 | − 2.48·2-s + 3.78·3-s − 1.83·4-s − 5·5-s − 9.39·6-s + 1.80·7-s + 24.4·8-s − 12.6·9-s + 12.4·10-s − 11·11-s − 6.96·12-s − 79.4·13-s − 4.48·14-s − 18.9·15-s − 45.9·16-s − 138.·17-s + 31.4·18-s − 19·19-s + 9.19·20-s + 6.83·21-s + 27.3·22-s + 24.6·23-s + 92.4·24-s + 25·25-s + 197.·26-s − 150.·27-s − 3.31·28-s + ⋯ |
| L(s) = 1 | − 0.877·2-s + 0.728·3-s − 0.229·4-s − 0.447·5-s − 0.639·6-s + 0.0974·7-s + 1.07·8-s − 0.469·9-s + 0.392·10-s − 0.301·11-s − 0.167·12-s − 1.69·13-s − 0.0855·14-s − 0.325·15-s − 0.717·16-s − 1.98·17-s + 0.411·18-s − 0.229·19-s + 0.102·20-s + 0.0710·21-s + 0.264·22-s + 0.223·23-s + 0.786·24-s + 0.200·25-s + 1.48·26-s − 1.07·27-s − 0.0224·28-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
| \(L(2)\) |
\(\approx\) |
\(0.5742020251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5742020251\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 2.48T + 8T^{2} \) |
| 3 | \( 1 - 3.78T + 27T^{2} \) |
| 7 | \( 1 - 1.80T + 343T^{2} \) |
| 13 | \( 1 + 79.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 138.T + 4.91e3T^{2} \) |
| 23 | \( 1 - 24.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 198.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 189.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 336.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 241.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 254.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 515.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 399.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 224.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 700.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 395.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 716.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.02e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 551.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 335.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 423.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 619.T + 9.12e5T^{2} \) |
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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
−9.371475484219160189634826224391, −8.684911048591851792353937832023, −8.103635249862337452378018871629, −7.40857713398836511404878309894, −6.45808392035562476927294132725, −4.84971512552532270722688850612, −4.43292877993059459293494850075, −2.92001961100762977304322181748, −2.10348813112737394288273134295, −0.42010672676448809230809156200,
0.42010672676448809230809156200, 2.10348813112737394288273134295, 2.92001961100762977304322181748, 4.43292877993059459293494850075, 4.84971512552532270722688850612, 6.45808392035562476927294132725, 7.40857713398836511404878309894, 8.103635249862337452378018871629, 8.684911048591851792353937832023, 9.371475484219160189634826224391
