| L(s) = 1 | + (−0.366 + 1.36i)2-s + (−1.73 − i)4-s + i·5-s + (−1.73 − 2i)7-s + (2 − 1.99i)8-s + (−1.36 − 0.366i)10-s + 0.267i·11-s − 0.464i·13-s + (3.36 − 1.63i)14-s + (1.99 + 3.46i)16-s + 6.46i·17-s + 6·19-s + (1 − 1.73i)20-s + (−0.366 − 0.0980i)22-s + 1.46i·23-s + ⋯ |
| L(s) = 1 | + (−0.258 + 0.965i)2-s + (−0.866 − 0.5i)4-s + 0.447i·5-s + (−0.654 − 0.755i)7-s + (0.707 − 0.707i)8-s + (−0.431 − 0.115i)10-s + 0.0807i·11-s − 0.128i·13-s + (0.899 − 0.436i)14-s + (0.499 + 0.866i)16-s + 1.56i·17-s + 1.37·19-s + (0.223 − 0.387i)20-s + (−0.0780 − 0.0209i)22-s + 0.305i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5243893838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5243893838\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.366 - 1.36i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (1.73 + 2i)T \) |
| good | 11 | \( 1 - 0.267iT - 11T^{2} \) |
| 13 | \( 1 + 0.464iT - 13T^{2} \) |
| 17 | \( 1 - 6.46iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 1.46iT - 23T^{2} \) |
| 29 | \( 1 + 7.92T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 9.46T + 37T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + 1.73T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 - 9.46iT - 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 - 7.46iT - 71T^{2} \) |
| 73 | \( 1 - 12.9iT - 73T^{2} \) |
| 79 | \( 1 + 14.6iT - 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 + 2.53iT - 89T^{2} \) |
| 97 | \( 1 + 13.3iT - 97T^{2} \) |
| 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
−10.03133004798903118032824318717, −9.283631045370898838948095822430, −8.387220638509648506513552664043, −7.39027138374309329729732738294, −7.07539159001562766655937554336, −6.01339771097742783838836994610, −5.38921778052880649527945137046, −4.03542945526132374422165792274, −3.41177078122350229259841738013, −1.48668169282314615412898482418,
0.24845776369713619113655862292, 1.79544813769263076766101732797, 2.93760759013415194677871181121, 3.70188111085925873953702937822, 5.05342224264864557462213353156, 5.50787550778464632882362838380, 6.96856772958390424125669121375, 7.77938951883100313070169905724, 8.856141463490153417501524945981, 9.352436791081230917933445067520
