| L(s) = 1 | − 4.44i·5-s − 4.87·7-s − 3.46i·11-s − 14.7·25-s − 5.34i·29-s − 10.5·31-s + 21.7i·35-s + 16.7·49-s − 12.4i·53-s − 15.4·55-s + 11.3i·59-s + 9.79·73-s + 16.8i·77-s − 3.32·79-s − 17.3i·83-s + ⋯ |
| L(s) = 1 | − 1.98i·5-s − 1.84·7-s − 1.04i·11-s − 2.95·25-s − 0.993i·29-s − 1.89·31-s + 3.66i·35-s + 2.39·49-s − 1.71i·53-s − 2.07·55-s + 1.47i·59-s + 1.14·73-s + 1.92i·77-s − 0.373·79-s − 1.90i·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2712474044\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2712474044\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 4.44iT - 5T^{2} \) |
| 7 | \( 1 + 4.87T + 7T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 5.34iT - 29T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 12.4iT - 53T^{2} \) |
| 59 | \( 1 - 11.3iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 9.79T + 73T^{2} \) |
| 79 | \( 1 + 3.32T + 79T^{2} \) |
| 83 | \( 1 + 17.3iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
| 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
−7.924398512962470235987677559272, −7.05707390645608814911149508565, −5.96420546501929469077153160756, −5.78847817271825673353150673707, −4.83597236100859498266028119682, −3.88240821023650305652315524442, −3.36876295914023434856505420420, −2.08731612034169401707311823394, −0.801052942294954626888092120736, −0.096176567685100685597859390153,
1.99956596120062715649452909718, 2.81453533290201097652428949545, 3.42807037110476589640989581016, 4.01999942347285650344583474377, 5.43246628375978665059861108721, 6.19724321945393169567066019662, 6.84402305988326198468270239995, 7.09754618565836936304254867611, 7.82450977482689565972218305457, 9.215362143685097671326049692982
