| L(s) = 1 | + (1.58 − 1.58i)5-s − i·7-s + 3.16·11-s + 3i·13-s − 6.32i·17-s − 3·19-s − 3.16i·23-s − 5.00i·25-s + 9.48·29-s − 2·31-s + (−1.58 − 1.58i)35-s − i·37-s + 3.16·41-s + 10i·43-s + 6.32i·47-s + ⋯ |
| L(s) = 1 | + (0.707 − 0.707i)5-s − 0.377i·7-s + 0.953·11-s + 0.832i·13-s − 1.53i·17-s − 0.688·19-s − 0.659i·23-s − 1.00i·25-s + 1.76·29-s − 0.359·31-s + (−0.267 − 0.267i)35-s − 0.164i·37-s + 0.493·41-s + 1.52i·43-s + 0.922i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.53725 - 0.636753i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.53725 - 0.636753i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.58 + 1.58i)T \) |
| good | 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 - 3.16T + 11T^{2} \) |
| 13 | \( 1 - 3iT - 13T^{2} \) |
| 17 | \( 1 + 6.32iT - 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 + 3.16iT - 23T^{2} \) |
| 29 | \( 1 - 9.48T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + iT - 37T^{2} \) |
| 41 | \( 1 - 3.16T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 - 6.32iT - 47T^{2} \) |
| 53 | \( 1 + 9.48iT - 53T^{2} \) |
| 59 | \( 1 + 6.32T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 + 11iT - 67T^{2} \) |
| 71 | \( 1 + 9.48T + 71T^{2} \) |
| 73 | \( 1 - 13iT - 73T^{2} \) |
| 79 | \( 1 - 3T + 79T^{2} \) |
| 83 | \( 1 - 15.8iT - 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - iT - 97T^{2} \) |
| 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
−10.66394558388108740158397214785, −9.602449061266310040709086096207, −9.138971882917032325124576494721, −8.180086797143006838135397086891, −6.87961662612998868504155277168, −6.25582017012099008313409589401, −4.91354576492017427015098996358, −4.21871151075220772968317566886, −2.55891320612509104968977451475, −1.11141292196279116917135978453,
1.68221706706578788417056345830, 2.98148059706059539763732947131, 4.14142567463601568673380914962, 5.64314905237052058772885131642, 6.23697560124964681076913275600, 7.19536141654616463916172647649, 8.402305423639278671659480762568, 9.133847116070031301149343471843, 10.30376404642834083141339154613, 10.61786359226985500153371548654
