| L(s) = 1 | − 1.25·2-s − 0.414·4-s + 1.25·5-s + 3.04·8-s − 1.58·10-s − 4.82·11-s − 1.17·13-s − 3·16-s + 4.29·17-s − 1.58·19-s − 0.521·20-s + 6.07·22-s + 6.60·23-s − 3.41·25-s + 1.47·26-s − 5.03·29-s − 10.6·31-s − 2.30·32-s − 5.41·34-s − 5.24·37-s + 1.99·38-s + 3.82·40-s + 0.521·41-s + 7.07·43-s + 1.99·44-s − 8.31·46-s + 11.4·47-s + ⋯ |
| L(s) = 1 | − 0.890·2-s − 0.207·4-s + 0.563·5-s + 1.07·8-s − 0.501·10-s − 1.45·11-s − 0.324·13-s − 0.750·16-s + 1.04·17-s − 0.363·19-s − 0.116·20-s + 1.29·22-s + 1.37·23-s − 0.682·25-s + 0.289·26-s − 0.935·29-s − 1.91·31-s − 0.407·32-s − 0.928·34-s − 0.861·37-s + 0.323·38-s + 0.605·40-s + 0.0814·41-s + 1.07·43-s + 0.301·44-s − 1.22·46-s + 1.66·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.25T + 2T^{2} \) |
| 5 | \( 1 - 1.25T + 5T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 + 1.17T + 13T^{2} \) |
| 17 | \( 1 - 4.29T + 17T^{2} \) |
| 19 | \( 1 + 1.58T + 19T^{2} \) |
| 23 | \( 1 - 6.60T + 23T^{2} \) |
| 29 | \( 1 + 5.03T + 29T^{2} \) |
| 31 | \( 1 + 10.6T + 31T^{2} \) |
| 37 | \( 1 + 5.24T + 37T^{2} \) |
| 41 | \( 1 - 0.521T + 41T^{2} \) |
| 43 | \( 1 - 7.07T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + 1.04T + 53T^{2} \) |
| 59 | \( 1 + 1.78T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 - 8.81T + 71T^{2} \) |
| 73 | \( 1 + 8.58T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 + 17.7T + 89T^{2} \) |
| 97 | \( 1 + 9.65T + 97T^{2} \) |
| 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
−9.277615219468600545939695987694, −8.561486358240621919374222942013, −7.55936304287013418937277825675, −7.26366739397076294346842574272, −5.62863390582758952854802152698, −5.28006968177684410322761131155, −4.03308990378162230604245431236, −2.71889605492296418850657063151, −1.55390764973579340074370716357, 0,
1.55390764973579340074370716357, 2.71889605492296418850657063151, 4.03308990378162230604245431236, 5.28006968177684410322761131155, 5.62863390582758952854802152698, 7.26366739397076294346842574272, 7.55936304287013418937277825675, 8.561486358240621919374222942013, 9.277615219468600545939695987694
