| L(s) = 1 | − 2-s − 1.75·3-s + 4-s − 3.88·5-s + 1.75·6-s + 3.77·7-s − 8-s + 0.0773·9-s + 3.88·10-s − 0.891·11-s − 1.75·12-s + 0.270·13-s − 3.77·14-s + 6.82·15-s + 16-s + 3.78·17-s − 0.0773·18-s + 6.97·19-s − 3.88·20-s − 6.62·21-s + 0.891·22-s + 1.59·23-s + 1.75·24-s + 10.1·25-s − 0.270·26-s + 5.12·27-s + 3.77·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.01·3-s + 0.5·4-s − 1.73·5-s + 0.716·6-s + 1.42·7-s − 0.353·8-s + 0.0257·9-s + 1.22·10-s − 0.268·11-s − 0.506·12-s + 0.0749·13-s − 1.00·14-s + 1.76·15-s + 0.250·16-s + 0.918·17-s − 0.0182·18-s + 1.59·19-s − 0.869·20-s − 1.44·21-s + 0.190·22-s + 0.332·23-s + 0.358·24-s + 2.02·25-s − 0.0529·26-s + 0.986·27-s + 0.713·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9143917382\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9143917382\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 4021 | \( 1+O(T) \) |
| good | 3 | \( 1 + 1.75T + 3T^{2} \) |
| 5 | \( 1 + 3.88T + 5T^{2} \) |
| 7 | \( 1 - 3.77T + 7T^{2} \) |
| 11 | \( 1 + 0.891T + 11T^{2} \) |
| 13 | \( 1 - 0.270T + 13T^{2} \) |
| 17 | \( 1 - 3.78T + 17T^{2} \) |
| 19 | \( 1 - 6.97T + 19T^{2} \) |
| 23 | \( 1 - 1.59T + 23T^{2} \) |
| 29 | \( 1 - 5.23T + 29T^{2} \) |
| 31 | \( 1 - 5.29T + 31T^{2} \) |
| 37 | \( 1 - 2.71T + 37T^{2} \) |
| 41 | \( 1 + 6.00T + 41T^{2} \) |
| 43 | \( 1 - 5.68T + 43T^{2} \) |
| 47 | \( 1 + 3.50T + 47T^{2} \) |
| 53 | \( 1 - 7.83T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 2.93T + 61T^{2} \) |
| 67 | \( 1 - 6.65T + 67T^{2} \) |
| 71 | \( 1 + 7.44T + 71T^{2} \) |
| 73 | \( 1 + 0.533T + 73T^{2} \) |
| 79 | \( 1 - 3.23T + 79T^{2} \) |
| 83 | \( 1 + 9.69T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 - 4.42T + 97T^{2} \) |
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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
−7.944127353504297416622878228942, −7.31646027354925137700416243203, −6.69632273180852357178309528075, −5.60055698945457534977044786062, −5.08625026592853102617435932186, −4.46999092575146270768366355828, −3.49914702916980970584227460499, −2.69380585152815482890738767538, −1.15732663174261107662154871631, −0.69346078885348720530717484033,
0.69346078885348720530717484033, 1.15732663174261107662154871631, 2.69380585152815482890738767538, 3.49914702916980970584227460499, 4.46999092575146270768366355828, 5.08625026592853102617435932186, 5.60055698945457534977044786062, 6.69632273180852357178309528075, 7.31646027354925137700416243203, 7.944127353504297416622878228942
