| L(s) = 1 | + i·2-s − 4-s + i·7-s − i·8-s + 4·11-s + 6i·13-s − 14-s + 16-s + 2i·17-s + 4·19-s + 4i·22-s − 8i·23-s − 6·26-s − i·28-s − 2·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.377i·7-s − 0.353i·8-s + 1.20·11-s + 1.66i·13-s − 0.267·14-s + 0.250·16-s + 0.485i·17-s + 0.917·19-s + 0.852i·22-s − 1.66i·23-s − 1.17·26-s − 0.188i·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.794713349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.794713349\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
| 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
−8.825662352621021287999239670911, −8.310515878353982249337371850513, −7.24146641322850673576644518077, −6.59696200645346197085099445930, −6.18226285667188940045780795352, −5.10187447989534779390674500199, −4.32895092309883762214038376186, −3.65610088890848707296211559053, −2.31982376803787028489023508474, −1.18213511157640488656663341229,
0.65489759606315591206840315201, 1.56770016999091544996877878190, 2.91013921001833239425327250775, 3.54646280687156728472762582249, 4.34336573609305764327316516822, 5.44936849187633603931885850179, 5.88122106670547835414241782326, 7.24352896766114055981756508196, 7.58708883784009702265886652514, 8.575688041178342675376580720894
