| L(s) = 1 | + 3.52i·5-s − 4.72i·7-s + (−0.146 + 3.31i)11-s − 2.10i·13-s + 4.68·17-s − 0.206i·19-s − 4.52i·23-s − 7.39·25-s − 7.66·29-s − 2.97·31-s + 16.6·35-s − 6.97·37-s − 5.27·41-s − 5.86i·43-s + 7.76i·47-s + ⋯ |
| L(s) = 1 | + 1.57i·5-s − 1.78i·7-s + (−0.0441 + 0.999i)11-s − 0.584i·13-s + 1.13·17-s − 0.0474i·19-s − 0.942i·23-s − 1.47·25-s − 1.42·29-s − 0.534·31-s + 2.81·35-s − 1.14·37-s − 0.823·41-s − 0.894i·43-s + 1.13i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 + 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2895002656\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2895002656\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (0.146 - 3.31i)T \) |
| good | 5 | \( 1 - 3.52iT - 5T^{2} \) |
| 7 | \( 1 + 4.72iT - 7T^{2} \) |
| 13 | \( 1 + 2.10iT - 13T^{2} \) |
| 17 | \( 1 - 4.68T + 17T^{2} \) |
| 19 | \( 1 + 0.206iT - 19T^{2} \) |
| 23 | \( 1 + 4.52iT - 23T^{2} \) |
| 29 | \( 1 + 7.66T + 29T^{2} \) |
| 31 | \( 1 + 2.97T + 31T^{2} \) |
| 37 | \( 1 + 6.97T + 37T^{2} \) |
| 41 | \( 1 + 5.27T + 41T^{2} \) |
| 43 | \( 1 + 5.86iT - 43T^{2} \) |
| 47 | \( 1 - 7.76iT - 47T^{2} \) |
| 53 | \( 1 - 6.34iT - 53T^{2} \) |
| 59 | \( 1 - 2.41iT - 59T^{2} \) |
| 61 | \( 1 + 14.3iT - 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 + 2.10iT - 71T^{2} \) |
| 73 | \( 1 - 13.1iT - 73T^{2} \) |
| 79 | \( 1 + 12.7iT - 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 4.24iT - 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
| 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
−7.57932863508684289109263076911, −7.02940117147005306809840721188, −6.68975610305363070253683924767, −5.68131798817636742987052155580, −4.79678195303918273678552806457, −3.78306566475457900204799478299, −3.49376721564221713321602758995, −2.48475984213997707066165008384, −1.43493314705180598595638019590, −0.07064350362245903598432020322,
1.33696788864502401530829111240, 2.01670488741429505754183125293, 3.19114753819138758898047062329, 3.90406279094941980568594966272, 5.08248910442306137144931588266, 5.50489634547235010015186792143, 5.76561523000226768951531045484, 6.87966938899113675316362286099, 7.951382771823083994561745806072, 8.424489726006587074286307314528
