| L(s) = 1 | − 4.01·2-s + 7.50·3-s + 8.09·4-s − 10.1·5-s − 30.1·6-s + 25.8·7-s − 0.391·8-s + 29.3·9-s + 40.8·10-s − 2.06·11-s + 60.7·12-s + 0.124·13-s − 103.·14-s − 76.4·15-s − 63.2·16-s − 86.9·17-s − 117.·18-s − 135.·19-s − 82.4·20-s + 194.·21-s + 8.30·22-s + 23·23-s − 2.94·24-s − 21.2·25-s − 0.500·26-s + 17.5·27-s + 209.·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.44·3-s + 1.01·4-s − 0.910·5-s − 2.04·6-s + 1.39·7-s − 0.0173·8-s + 1.08·9-s + 1.29·10-s − 0.0567·11-s + 1.46·12-s + 0.00266·13-s − 1.98·14-s − 1.31·15-s − 0.987·16-s − 1.24·17-s − 1.54·18-s − 1.63·19-s − 0.921·20-s + 2.01·21-s + 0.0804·22-s + 0.208·23-s − 0.0250·24-s − 0.170·25-s − 0.00377·26-s + 0.125·27-s + 1.41·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - 23T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 + 4.01T + 8T^{2} \) |
| 3 | \( 1 - 7.50T + 27T^{2} \) |
| 5 | \( 1 + 10.1T + 125T^{2} \) |
| 7 | \( 1 - 25.8T + 343T^{2} \) |
| 11 | \( 1 + 2.06T + 1.33e3T^{2} \) |
| 13 | \( 1 - 0.124T + 2.19e3T^{2} \) |
| 17 | \( 1 + 86.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 135.T + 6.85e3T^{2} \) |
| 31 | \( 1 - 144.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 202.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 309.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 105.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 81.2T + 1.03e5T^{2} \) |
| 53 | \( 1 + 235.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 445.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 583.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 476.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 679.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 116.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 589.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 187.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.39e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.03e3T + 9.12e5T^{2} \) |
| 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
−9.305329229402125748627514802349, −8.477818312543219438253907438709, −8.350793406055715884293069509298, −7.61384748367169549199075930565, −6.73844929357369703841654763805, −4.70986014300832258135902181683, −3.96201833594761428425139539188, −2.41160447605670532209129564433, −1.63630133899380000294530563117, 0,
1.63630133899380000294530563117, 2.41160447605670532209129564433, 3.96201833594761428425139539188, 4.70986014300832258135902181683, 6.73844929357369703841654763805, 7.61384748367169549199075930565, 8.350793406055715884293069509298, 8.477818312543219438253907438709, 9.305329229402125748627514802349
