| L(s) = 1 | − 0.0907·2-s − 3.16·3-s − 1.99·4-s − 2.27·5-s + 0.287·6-s − 4.87·7-s + 0.362·8-s + 7.03·9-s + 0.206·10-s + 1.57·11-s + 6.31·12-s + 3.90·13-s + 0.442·14-s + 7.20·15-s + 3.95·16-s + 5.23·17-s − 0.638·18-s + 19-s + 4.53·20-s + 15.4·21-s − 0.143·22-s + 7.94·23-s − 1.14·24-s + 0.178·25-s − 0.354·26-s − 12.7·27-s + 9.71·28-s + ⋯ |
| L(s) = 1 | − 0.0641·2-s − 1.82·3-s − 0.995·4-s − 1.01·5-s + 0.117·6-s − 1.84·7-s + 0.128·8-s + 2.34·9-s + 0.0653·10-s + 0.475·11-s + 1.82·12-s + 1.08·13-s + 0.118·14-s + 1.86·15-s + 0.987·16-s + 1.27·17-s − 0.150·18-s + 0.229·19-s + 1.01·20-s + 3.37·21-s − 0.0305·22-s + 1.65·23-s − 0.234·24-s + 0.0356·25-s − 0.0695·26-s − 2.46·27-s + 1.83·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4721824136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4721824136\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 - T \) |
| 317 | \( 1 + T \) |
| good | 2 | \( 1 + 0.0907T + 2T^{2} \) |
| 3 | \( 1 + 3.16T + 3T^{2} \) |
| 5 | \( 1 + 2.27T + 5T^{2} \) |
| 7 | \( 1 + 4.87T + 7T^{2} \) |
| 11 | \( 1 - 1.57T + 11T^{2} \) |
| 13 | \( 1 - 3.90T + 13T^{2} \) |
| 17 | \( 1 - 5.23T + 17T^{2} \) |
| 23 | \( 1 - 7.94T + 23T^{2} \) |
| 29 | \( 1 + 6.86T + 29T^{2} \) |
| 31 | \( 1 + 0.267T + 31T^{2} \) |
| 37 | \( 1 - 7.84T + 37T^{2} \) |
| 41 | \( 1 - 9.28T + 41T^{2} \) |
| 43 | \( 1 + 9.11T + 43T^{2} \) |
| 47 | \( 1 - 9.30T + 47T^{2} \) |
| 53 | \( 1 + 6.52T + 53T^{2} \) |
| 59 | \( 1 + 6.66T + 59T^{2} \) |
| 61 | \( 1 + 2.74T + 61T^{2} \) |
| 67 | \( 1 - 6.30T + 67T^{2} \) |
| 71 | \( 1 - 9.28T + 71T^{2} \) |
| 73 | \( 1 + 6.68T + 73T^{2} \) |
| 79 | \( 1 - 16.2T + 79T^{2} \) |
| 83 | \( 1 - 1.60T + 83T^{2} \) |
| 89 | \( 1 - 2.83T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.82428777693165274802104129016, −7.30441679687923665164014345093, −6.43534700711396389063222158664, −5.96406506740511901301273496532, −5.34031171178060415744880120003, −4.45732401844053150994627657095, −3.69383446148350475941694742658, −3.35345197082868415082719152811, −1.05105539447046402360736543770, −0.53600771385432595781170238994,
0.53600771385432595781170238994, 1.05105539447046402360736543770, 3.35345197082868415082719152811, 3.69383446148350475941694742658, 4.45732401844053150994627657095, 5.34031171178060415744880120003, 5.96406506740511901301273496532, 6.43534700711396389063222158664, 7.30441679687923665164014345093, 7.82428777693165274802104129016
