| L(s) = 1 | − 1.11·5-s + (−1.37 + 2.26i)7-s + 0.812·11-s − 2·13-s + 3.55i·17-s + 4.52i·19-s − 3.63i·23-s − 3.75·25-s − 8.95i·29-s − 5.08·31-s + (1.53 − 2.52i)35-s − 0.718i·37-s + 10.4i·41-s − 1.11·43-s + 8.55·47-s + ⋯ |
| L(s) = 1 | − 0.498·5-s + (−0.518 + 0.854i)7-s + 0.245·11-s − 0.554·13-s + 0.862i·17-s + 1.03i·19-s − 0.758i·23-s − 0.751·25-s − 1.66i·29-s − 0.914·31-s + (0.258 − 0.426i)35-s − 0.118i·37-s + 1.63i·41-s − 0.170·43-s + 1.24·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.279 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4383073617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4383073617\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.37 - 2.26i)T \) |
| good | 5 | \( 1 + 1.11T + 5T^{2} \) |
| 11 | \( 1 - 0.812T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 3.55iT - 17T^{2} \) |
| 19 | \( 1 - 4.52iT - 19T^{2} \) |
| 23 | \( 1 + 3.63iT - 23T^{2} \) |
| 29 | \( 1 + 8.95iT - 29T^{2} \) |
| 31 | \( 1 + 5.08T + 31T^{2} \) |
| 37 | \( 1 + 0.718iT - 37T^{2} \) |
| 41 | \( 1 - 10.4iT - 41T^{2} \) |
| 43 | \( 1 + 1.11T + 43T^{2} \) |
| 47 | \( 1 - 8.55T + 47T^{2} \) |
| 53 | \( 1 + 5.08iT - 53T^{2} \) |
| 59 | \( 1 + 2.23iT - 59T^{2} \) |
| 61 | \( 1 + 5.27T + 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 + 5.87iT - 71T^{2} \) |
| 73 | \( 1 - 3.86iT - 73T^{2} \) |
| 79 | \( 1 - 1.70iT - 79T^{2} \) |
| 83 | \( 1 + 7.27iT - 83T^{2} \) |
| 89 | \( 1 + 3.37iT - 89T^{2} \) |
| 97 | \( 1 + 8.55iT - 97T^{2} \) |
| 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
−8.125161046785251134606611166817, −7.71493488428594758852230754560, −6.60848685918165014670968362058, −6.06607647368307249696584459099, −5.35737982823051738539246412398, −4.26828187803019365757215980890, −3.68143534115498534905909346365, −2.64657026959000069683243882718, −1.78437354174787926943007068905, −0.14422389759729768822488020222,
0.985269156130291950341533152830, 2.35642398642689717788434069491, 3.38311308517913439071447340374, 3.96845674587013688701698013371, 4.90022861821808473972720318264, 5.59323561099948001880025325771, 6.74747270317979117617644962884, 7.23596680440444497451999476484, 7.64011400675157286267868070019, 8.804772760548539150640125838102
