| L(s) = 1 | + 2-s + 0.630·3-s + 4-s + 1.86·5-s + 0.630·6-s + 3.14·7-s + 8-s − 2.60·9-s + 1.86·10-s − 0.744·11-s + 0.630·12-s + 1.35·13-s + 3.14·14-s + 1.17·15-s + 16-s + 2.12·17-s − 2.60·18-s + 5.73·19-s + 1.86·20-s + 1.98·21-s − 0.744·22-s + 4.64·23-s + 0.630·24-s − 1.51·25-s + 1.35·26-s − 3.53·27-s + 3.14·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.363·3-s + 0.5·4-s + 0.834·5-s + 0.257·6-s + 1.19·7-s + 0.353·8-s − 0.867·9-s + 0.590·10-s − 0.224·11-s + 0.181·12-s + 0.375·13-s + 0.841·14-s + 0.303·15-s + 0.250·16-s + 0.515·17-s − 0.613·18-s + 1.31·19-s + 0.417·20-s + 0.433·21-s − 0.158·22-s + 0.968·23-s + 0.128·24-s − 0.303·25-s + 0.265·26-s − 0.679·27-s + 0.595·28-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
| \(L(1)\) |
\(\approx\) |
\(4.716399640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.716399640\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.630T + 3T^{2} \) |
| 5 | \( 1 - 1.86T + 5T^{2} \) |
| 7 | \( 1 - 3.14T + 7T^{2} \) |
| 11 | \( 1 + 0.744T + 11T^{2} \) |
| 13 | \( 1 - 1.35T + 13T^{2} \) |
| 17 | \( 1 - 2.12T + 17T^{2} \) |
| 19 | \( 1 - 5.73T + 19T^{2} \) |
| 23 | \( 1 - 4.64T + 23T^{2} \) |
| 29 | \( 1 + 1.91T + 29T^{2} \) |
| 31 | \( 1 + 2.00T + 31T^{2} \) |
| 37 | \( 1 - 1.98T + 37T^{2} \) |
| 41 | \( 1 - 7.72T + 41T^{2} \) |
| 43 | \( 1 + 12.5T + 43T^{2} \) |
| 47 | \( 1 - 3.43T + 47T^{2} \) |
| 53 | \( 1 + 7.21T + 53T^{2} \) |
| 59 | \( 1 + 1.76T + 59T^{2} \) |
| 61 | \( 1 - 2.62T + 61T^{2} \) |
| 67 | \( 1 + 2.75T + 67T^{2} \) |
| 71 | \( 1 - 9.38T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 3.40T + 89T^{2} \) |
| 97 | \( 1 - 3.15T + 97T^{2} \) |
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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
−8.197959108047189978324468989345, −7.86976254030205238426954438418, −6.92960833686826936471945370079, −5.97604584274677320786167178274, −5.34984730435394302678843485589, −4.96198819131388264661444385118, −3.74774400482589861001004181920, −2.97641744408791096452682475281, −2.11117147037667089102500213183, −1.22172932828294322989106799504,
1.22172932828294322989106799504, 2.11117147037667089102500213183, 2.97641744408791096452682475281, 3.74774400482589861001004181920, 4.96198819131388264661444385118, 5.34984730435394302678843485589, 5.97604584274677320786167178274, 6.92960833686826936471945370079, 7.86976254030205238426954438418, 8.197959108047189978324468989345
