| L(s) = 1 | − 2-s + 1.56·3-s + 4-s − 5-s − 1.56·6-s − 7-s − 8-s − 0.561·9-s + 10-s + 2·11-s + 1.56·12-s − 5.56·13-s + 14-s − 1.56·15-s + 16-s − 17-s + 0.561·18-s + 3.56·19-s − 20-s − 1.56·21-s − 2·22-s − 3.12·23-s − 1.56·24-s + 25-s + 5.56·26-s − 5.56·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.901·3-s + 0.5·4-s − 0.447·5-s − 0.637·6-s − 0.377·7-s − 0.353·8-s − 0.187·9-s + 0.316·10-s + 0.603·11-s + 0.450·12-s − 1.54·13-s + 0.267·14-s − 0.403·15-s + 0.250·16-s − 0.242·17-s + 0.132·18-s + 0.817·19-s − 0.223·20-s − 0.340·21-s − 0.426·22-s − 0.651·23-s − 0.318·24-s + 0.200·25-s + 1.09·26-s − 1.07·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 5.56T + 13T^{2} \) |
| 19 | \( 1 - 3.56T + 19T^{2} \) |
| 23 | \( 1 + 3.12T + 23T^{2} \) |
| 29 | \( 1 + 1.56T + 29T^{2} \) |
| 31 | \( 1 + 8.68T + 31T^{2} \) |
| 37 | \( 1 - 3.12T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 2.43T + 47T^{2} \) |
| 53 | \( 1 + 0.438T + 53T^{2} \) |
| 59 | \( 1 + 14.6T + 59T^{2} \) |
| 61 | \( 1 + 4.43T + 61T^{2} \) |
| 67 | \( 1 - 6.24T + 67T^{2} \) |
| 71 | \( 1 - 6.43T + 71T^{2} \) |
| 73 | \( 1 + 4.43T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 2.68T + 97T^{2} \) |
| 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
−9.430494118105506407119418386457, −8.604277038773813775104761511625, −7.71385327201993143404155243858, −7.27819061717395179840091709859, −6.21095265235112476672909097048, −5.04331127141537554466308599641, −3.75772783549148859762998830016, −2.93420196328833310057657339218, −1.89492050422220215976475606174, 0,
1.89492050422220215976475606174, 2.93420196328833310057657339218, 3.75772783549148859762998830016, 5.04331127141537554466308599641, 6.21095265235112476672909097048, 7.27819061717395179840091709859, 7.71385327201993143404155243858, 8.604277038773813775104761511625, 9.430494118105506407119418386457
