| L(s) = 1 | − 2i·2-s − 4·4-s + 8i·7-s + 8i·8-s − 18·11-s − 8i·13-s + 16·14-s + 16·16-s − 15i·17-s − 23·19-s + 36i·22-s + 63i·23-s − 16·26-s − 32i·28-s + 156·29-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.431i·7-s + 0.353i·8-s − 0.493·11-s − 0.170i·13-s + 0.305·14-s + 0.250·16-s − 0.214i·17-s − 0.277·19-s + 0.348i·22-s + 0.571i·23-s − 0.120·26-s − 0.215i·28-s + 0.998·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.294378548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.294378548\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 8iT - 343T^{2} \) |
| 11 | \( 1 + 18T + 1.33e3T^{2} \) |
| 13 | \( 1 + 8iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 15iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 23T + 6.85e3T^{2} \) |
| 23 | \( 1 - 63iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 156T + 2.43e4T^{2} \) |
| 31 | \( 1 + 85T + 2.97e4T^{2} \) |
| 37 | \( 1 - 74iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 246T + 6.89e4T^{2} \) |
| 43 | \( 1 - 190iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 288iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 177iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 792T + 2.05e5T^{2} \) |
| 61 | \( 1 + 907T + 2.26e5T^{2} \) |
| 67 | \( 1 + 322iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 270T + 3.57e5T^{2} \) |
| 73 | \( 1 + 254iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.12e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 771iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 198T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.19e3iT - 9.12e5T^{2} \) |
| 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
−9.020044504121937035643701172503, −8.353195570458416279327667275473, −7.49155249123502089064396012275, −6.42862237274745087268140635416, −5.43890016358136146272405227830, −4.72326863637637469267147920416, −3.58054102821284159636179980814, −2.70578328638284405629365032981, −1.70784678278531129836639100425, −0.36897391655661560704941339429,
0.889540199155425317831500727959, 2.36217544100674672555370484987, 3.62930287688789995163077104701, 4.52675277112721968561828587942, 5.36677628565237915914982529739, 6.31849997047076611249966254867, 7.02802857816878580288347202770, 7.84753194929223053871921308921, 8.559179483849179370778482265961, 9.343993016173075581998105905121
