| L(s) = 1 | − 2.48·2-s + 1.66·3-s + 4.19·4-s + 0.253·5-s − 4.14·6-s − 2.12·7-s − 5.46·8-s − 0.230·9-s − 0.630·10-s − 4.54·11-s + 6.98·12-s − 13-s + 5.29·14-s + 0.421·15-s + 5.21·16-s − 4.33·17-s + 0.574·18-s + 2.44·19-s + 1.06·20-s − 3.54·21-s + 11.3·22-s − 0.598·23-s − 9.09·24-s − 4.93·25-s + 2.48·26-s − 5.37·27-s − 8.93·28-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 0.960·3-s + 2.09·4-s + 0.113·5-s − 1.69·6-s − 0.804·7-s − 1.93·8-s − 0.0769·9-s − 0.199·10-s − 1.37·11-s + 2.01·12-s − 0.277·13-s + 1.41·14-s + 0.108·15-s + 1.30·16-s − 1.05·17-s + 0.135·18-s + 0.561·19-s + 0.237·20-s − 0.772·21-s + 2.41·22-s − 0.124·23-s − 1.85·24-s − 0.987·25-s + 0.488·26-s − 1.03·27-s − 1.68·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3399206655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3399206655\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + T \) |
| 619 | \( 1 - T \) |
| good | 2 | \( 1 + 2.48T + 2T^{2} \) |
| 3 | \( 1 - 1.66T + 3T^{2} \) |
| 5 | \( 1 - 0.253T + 5T^{2} \) |
| 7 | \( 1 + 2.12T + 7T^{2} \) |
| 11 | \( 1 + 4.54T + 11T^{2} \) |
| 17 | \( 1 + 4.33T + 17T^{2} \) |
| 19 | \( 1 - 2.44T + 19T^{2} \) |
| 23 | \( 1 + 0.598T + 23T^{2} \) |
| 29 | \( 1 + 7.79T + 29T^{2} \) |
| 31 | \( 1 + 0.603T + 31T^{2} \) |
| 37 | \( 1 + 9.24T + 37T^{2} \) |
| 41 | \( 1 + 0.533T + 41T^{2} \) |
| 43 | \( 1 - 0.273T + 43T^{2} \) |
| 47 | \( 1 - 9.67T + 47T^{2} \) |
| 53 | \( 1 - 5.45T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 1.34T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 - 4.97T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 - 2.66T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−7.987471854872370368190992268733, −7.37435505225982510371009370502, −6.96007166799414185319612830491, −5.94193936001653937037125616062, −5.31721030131420970625921621007, −3.92465803934789499902026557516, −3.08461956032771752751924503862, −2.36937694820896943252184866562, −1.90092952179298608061066211908, −0.32942032914473001429934427836,
0.32942032914473001429934427836, 1.90092952179298608061066211908, 2.36937694820896943252184866562, 3.08461956032771752751924503862, 3.92465803934789499902026557516, 5.31721030131420970625921621007, 5.94193936001653937037125616062, 6.96007166799414185319612830491, 7.37435505225982510371009370502, 7.987471854872370368190992268733
