| L(s) = 1 | − 2-s − 3.15·3-s + 4-s − 2.19·5-s + 3.15·6-s − 1.72·7-s − 8-s + 6.93·9-s + 2.19·10-s − 0.439·11-s − 3.15·12-s − 3.50·13-s + 1.72·14-s + 6.92·15-s + 16-s + 4.60·17-s − 6.93·18-s + 3.98·19-s − 2.19·20-s + 5.44·21-s + 0.439·22-s + 0.0263·23-s + 3.15·24-s − 0.179·25-s + 3.50·26-s − 12.4·27-s − 1.72·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.81·3-s + 0.5·4-s − 0.981·5-s + 1.28·6-s − 0.652·7-s − 0.353·8-s + 2.31·9-s + 0.694·10-s − 0.132·11-s − 0.909·12-s − 0.971·13-s + 0.461·14-s + 1.78·15-s + 0.250·16-s + 1.11·17-s − 1.63·18-s + 0.914·19-s − 0.490·20-s + 1.18·21-s + 0.0936·22-s + 0.00549·23-s + 0.643·24-s − 0.0359·25-s + 0.686·26-s − 2.38·27-s − 0.326·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2362338654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2362338654\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 + 3.15T + 3T^{2} \) |
| 5 | \( 1 + 2.19T + 5T^{2} \) |
| 7 | \( 1 + 1.72T + 7T^{2} \) |
| 11 | \( 1 + 0.439T + 11T^{2} \) |
| 13 | \( 1 + 3.50T + 13T^{2} \) |
| 17 | \( 1 - 4.60T + 17T^{2} \) |
| 19 | \( 1 - 3.98T + 19T^{2} \) |
| 23 | \( 1 - 0.0263T + 23T^{2} \) |
| 29 | \( 1 - 2.19T + 29T^{2} \) |
| 31 | \( 1 - 5.76T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 - 1.19T + 41T^{2} \) |
| 43 | \( 1 + 7.29T + 43T^{2} \) |
| 47 | \( 1 + 1.93T + 47T^{2} \) |
| 53 | \( 1 + 3.98T + 53T^{2} \) |
| 59 | \( 1 + 9.95T + 59T^{2} \) |
| 61 | \( 1 - 7.51T + 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 - 4.29T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 - 9.98T + 79T^{2} \) |
| 83 | \( 1 - 6.44T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 4.96T + 97T^{2} \) |
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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
−8.180318012290449745342012712541, −7.63686590438516455389874583573, −6.88815235671698470230686625732, −6.45935923680553480457622830743, −5.40825423936574735526286080462, −5.00072417716051676821895994039, −3.91019124948419363604675954933, −3.00472678124350925139696292669, −1.40864506763689242583230628309, −0.36302929291464471354018871330,
0.36302929291464471354018871330, 1.40864506763689242583230628309, 3.00472678124350925139696292669, 3.91019124948419363604675954933, 5.00072417716051676821895994039, 5.40825423936574735526286080462, 6.45935923680553480457622830743, 6.88815235671698470230686625732, 7.63686590438516455389874583573, 8.180318012290449745342012712541
