| L(s) = 1 | + 2-s − 0.801·3-s + 4-s − 3.04·5-s − 0.801·6-s − 7-s + 8-s − 2.35·9-s − 3.04·10-s − 1.24·11-s − 0.801·12-s − 14-s + 2.44·15-s + 16-s − 1.13·17-s − 2.35·18-s + 6.69·19-s − 3.04·20-s + 0.801·21-s − 1.24·22-s − 2.93·23-s − 0.801·24-s + 4.29·25-s + 4.29·27-s − 28-s − 4.54·29-s + 2.44·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.462·3-s + 0.5·4-s − 1.36·5-s − 0.327·6-s − 0.377·7-s + 0.353·8-s − 0.785·9-s − 0.964·10-s − 0.375·11-s − 0.231·12-s − 0.267·14-s + 0.631·15-s + 0.250·16-s − 0.275·17-s − 0.555·18-s + 1.53·19-s − 0.681·20-s + 0.174·21-s − 0.265·22-s − 0.612·23-s − 0.163·24-s + 0.859·25-s + 0.826·27-s − 0.188·28-s − 0.843·29-s + 0.446·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.307002919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.307002919\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 0.801T + 3T^{2} \) |
| 5 | \( 1 + 3.04T + 5T^{2} \) |
| 11 | \( 1 + 1.24T + 11T^{2} \) |
| 17 | \( 1 + 1.13T + 17T^{2} \) |
| 19 | \( 1 - 6.69T + 19T^{2} \) |
| 23 | \( 1 + 2.93T + 23T^{2} \) |
| 29 | \( 1 + 4.54T + 29T^{2} \) |
| 31 | \( 1 + 1.26T + 31T^{2} \) |
| 37 | \( 1 - 9.44T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 + 7.89T + 43T^{2} \) |
| 47 | \( 1 - 6.18T + 47T^{2} \) |
| 53 | \( 1 - 0.405T + 53T^{2} \) |
| 59 | \( 1 - 3.07T + 59T^{2} \) |
| 61 | \( 1 - 7.12T + 61T^{2} \) |
| 67 | \( 1 + 9.00T + 67T^{2} \) |
| 71 | \( 1 + 2.02T + 71T^{2} \) |
| 73 | \( 1 - 16.8T + 73T^{2} \) |
| 79 | \( 1 - 16.3T + 79T^{2} \) |
| 83 | \( 1 + 4.93T + 83T^{2} \) |
| 89 | \( 1 - 2.51T + 89T^{2} \) |
| 97 | \( 1 + 8.04T + 97T^{2} \) |
| 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
−8.910501463030750907306386368707, −7.83683571391805489191070037880, −7.57366572163028341905736455192, −6.53500235615994553079807682967, −5.73522778857034344744748916912, −5.04785349472335484044784897120, −4.07704227736615626234234253296, −3.40020875293993581659968882287, −2.50498974873250465675792636204, −0.65161611358665726622005258345,
0.65161611358665726622005258345, 2.50498974873250465675792636204, 3.40020875293993581659968882287, 4.07704227736615626234234253296, 5.04785349472335484044784897120, 5.73522778857034344744748916912, 6.53500235615994553079807682967, 7.57366572163028341905736455192, 7.83683571391805489191070037880, 8.910501463030750907306386368707
