L(s) = 1 | + 5.51·2-s − 6.01·3-s + 22.4·4-s − 5·5-s − 33.1·6-s + 22.7·7-s + 79.8·8-s + 9.17·9-s − 27.5·10-s − 11·11-s − 135.·12-s + 8.83·13-s + 125.·14-s + 30.0·15-s + 261.·16-s + 122.·17-s + 50.6·18-s − 19·19-s − 112.·20-s − 136.·21-s − 60.7·22-s − 177.·23-s − 480.·24-s + 25·25-s + 48.7·26-s + 107.·27-s + 510.·28-s + ⋯ |
L(s) = 1 | + 1.95·2-s − 1.15·3-s + 2.80·4-s − 0.447·5-s − 2.25·6-s + 1.22·7-s + 3.52·8-s + 0.339·9-s − 0.872·10-s − 0.301·11-s − 3.25·12-s + 0.188·13-s + 2.39·14-s + 0.517·15-s + 4.07·16-s + 1.74·17-s + 0.663·18-s − 0.229·19-s − 1.25·20-s − 1.42·21-s − 0.588·22-s − 1.60·23-s − 4.08·24-s + 0.200·25-s + 0.368·26-s + 0.764·27-s + 3.44·28-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
\(L(2)\) |
\(\approx\) |
\(6.015619452\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.015619452\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 - 5.51T + 8T^{2} \) |
| 3 | \( 1 + 6.01T + 27T^{2} \) |
| 7 | \( 1 - 22.7T + 343T^{2} \) |
| 13 | \( 1 - 8.83T + 2.19e3T^{2} \) |
| 17 | \( 1 - 122.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 177.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 36.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 53.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 133.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 359.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 189.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 495.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 504.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 404.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 228.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 280.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 293.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 589.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 62.7T + 4.93e5T^{2} \) |
| 83 | \( 1 - 856.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 923.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 340.T + 9.12e5T^{2} \) |
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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
−10.25450844785821304651256632540, −8.163745113719053116460065447499, −7.61029044393862697250954741704, −6.59884173899158229597112165623, −5.62133508612956472918838073736, −5.37420924220678844970294295086, −4.42163777369142896752028678834, −3.66459749787791445696512778770, −2.34574059431109824287785504012, −1.10097748345952676146748523543,
1.10097748345952676146748523543, 2.34574059431109824287785504012, 3.66459749787791445696512778770, 4.42163777369142896752028678834, 5.37420924220678844970294295086, 5.62133508612956472918838073736, 6.59884173899158229597112165623, 7.61029044393862697250954741704, 8.163745113719053116460065447499, 10.25450844785821304651256632540