| L(s) = 1 | − 3-s + (2 − i)5-s − 2·7-s − 2·9-s + i·13-s + (−2 + i)15-s + (1 + 4i)17-s + 5·19-s + 2·21-s + 4·23-s + (3 − 4i)25-s + 5·27-s − 9i·29-s − 5i·31-s + (−4 + 2i)35-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + (0.894 − 0.447i)5-s − 0.755·7-s − 0.666·9-s + 0.277i·13-s + (−0.516 + 0.258i)15-s + (0.242 + 0.970i)17-s + 1.14·19-s + 0.436·21-s + 0.834·23-s + (0.600 − 0.800i)25-s + 0.962·27-s − 1.67i·29-s − 0.898i·31-s + (−0.676 + 0.338i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.311130245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.311130245\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2 + i)T \) |
| 17 | \( 1 + (-1 - 4i)T \) |
| good | 3 | \( 1 + T + 3T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 9iT - 29T^{2} \) |
| 31 | \( 1 + 5iT - 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 - 7iT - 47T^{2} \) |
| 53 | \( 1 - iT - 53T^{2} \) |
| 59 | \( 1 - 5T + 59T^{2} \) |
| 61 | \( 1 - 5iT - 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + 5iT - 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 + 16iT - 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 5T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
| 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
−9.506682464028333256541569513862, −8.880400947871359757080493417518, −7.908156716180165623176827967377, −6.81533792514714104397001818228, −5.92696653533164703105422229547, −5.63288626757168650063895875931, −4.51359525259187955391358877618, −3.30083867240891028181980960366, −2.17873130603948586183061320840, −0.69143557889800983066736928795,
1.10090480288933046383849803078, 2.81387034130292095790894214638, 3.26210231054712695366240630478, 5.15860967764037305402975591030, 5.35274239356481581580309430137, 6.57016161830703691350497535512, 6.87233244465100796331072583981, 8.088935498853679782822268152305, 9.221230012720767847284460250312, 9.625092942629649352348578182839
