| L(s) = 1 | + i·3-s − 9-s − 4·11-s − 2i·13-s − 2i·17-s − 4·19-s + 8i·23-s − i·27-s − 6·29-s − 8·31-s − 4i·33-s − 6i·37-s + 2·39-s − 6·41-s − 4i·43-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.333·9-s − 1.20·11-s − 0.554i·13-s − 0.485i·17-s − 0.917·19-s + 1.66i·23-s − 0.192i·27-s − 1.11·29-s − 1.43·31-s − 0.696i·33-s − 0.986i·37-s + 0.320·39-s − 0.937·41-s − 0.609i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
| 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
−9.442517922138744862066974570191, −8.681755263756113727855447818436, −7.73362334914342770735122976781, −7.10487299799159393557263403488, −5.58338788664374810301843404421, −5.39975766186818179612635171872, −4.09791195080334812116643820526, −3.21109854637721406124469186890, −2.04555220160031712880904490069, 0,
1.80509395194441917922238338027, 2.73704710677610881646288420532, 4.03478677402135871280048824257, 5.05554635793203841842230107205, 6.00872006837976967030667424282, 6.80311028026961086387750024953, 7.65405360595228567799146346241, 8.421824338907487617689049402103, 9.095569430131749971545586709374
