| L(s) = 1 | − 2-s − 0.130·3-s + 4-s − 1.89·5-s + 0.130·6-s − 2.41·7-s − 8-s − 2.98·9-s + 1.89·10-s − 4.01·11-s − 0.130·12-s + 3.30·13-s + 2.41·14-s + 0.246·15-s + 16-s + 1.40·17-s + 2.98·18-s − 3.27·19-s − 1.89·20-s + 0.314·21-s + 4.01·22-s − 6.52·23-s + 0.130·24-s − 1.42·25-s − 3.30·26-s + 0.781·27-s − 2.41·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.0754·3-s + 0.5·4-s − 0.845·5-s + 0.0533·6-s − 0.911·7-s − 0.353·8-s − 0.994·9-s + 0.597·10-s − 1.21·11-s − 0.0377·12-s + 0.917·13-s + 0.644·14-s + 0.0637·15-s + 0.250·16-s + 0.340·17-s + 0.703·18-s − 0.752·19-s − 0.422·20-s + 0.0687·21-s + 0.856·22-s − 1.35·23-s + 0.0266·24-s − 0.285·25-s − 0.648·26-s + 0.150·27-s − 0.455·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2067508792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2067508792\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2003 | \( 1 - T \) |
| good | 3 | \( 1 + 0.130T + 3T^{2} \) |
| 5 | \( 1 + 1.89T + 5T^{2} \) |
| 7 | \( 1 + 2.41T + 7T^{2} \) |
| 11 | \( 1 + 4.01T + 11T^{2} \) |
| 13 | \( 1 - 3.30T + 13T^{2} \) |
| 17 | \( 1 - 1.40T + 17T^{2} \) |
| 19 | \( 1 + 3.27T + 19T^{2} \) |
| 23 | \( 1 + 6.52T + 23T^{2} \) |
| 29 | \( 1 - 2.08T + 29T^{2} \) |
| 31 | \( 1 - 0.507T + 31T^{2} \) |
| 37 | \( 1 + 9.29T + 37T^{2} \) |
| 41 | \( 1 - 3.75T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 + 0.725T + 47T^{2} \) |
| 53 | \( 1 - 5.56T + 53T^{2} \) |
| 59 | \( 1 + 4.91T + 59T^{2} \) |
| 61 | \( 1 + 6.90T + 61T^{2} \) |
| 67 | \( 1 + 5.72T + 67T^{2} \) |
| 71 | \( 1 + 9.92T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 + 8.24T + 79T^{2} \) |
| 83 | \( 1 - 6.79T + 83T^{2} \) |
| 89 | \( 1 - 4.72T + 89T^{2} \) |
| 97 | \( 1 - 8.19T + 97T^{2} \) |
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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
−8.448955309397436160890451603229, −7.893493722731827239334901791606, −7.16300448487147978996677722918, −6.14861627759704065843517965766, −5.83692300167523447034448318662, −4.65562295308979433841769496594, −3.53105847019373879872033009508, −3.05332649748589302444147191786, −1.93183097871773518281863678786, −0.27170088575235167101172130518,
0.27170088575235167101172130518, 1.93183097871773518281863678786, 3.05332649748589302444147191786, 3.53105847019373879872033009508, 4.65562295308979433841769496594, 5.83692300167523447034448318662, 6.14861627759704065843517965766, 7.16300448487147978996677722918, 7.893493722731827239334901791606, 8.448955309397436160890451603229
