| L(s) = 1 | + i·2-s − i·3-s − 4-s − 1.58i·5-s + 6-s − i·8-s − 9-s + 1.58·10-s + (3.29 + 0.383i)11-s + i·12-s + 2.26·13-s − 1.58·15-s + 16-s − 2.99·17-s − i·18-s + 1.32·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.707i·5-s + 0.408·6-s − 0.353i·8-s − 0.333·9-s + 0.500·10-s + (0.993 + 0.115i)11-s + 0.288i·12-s + 0.629·13-s − 0.408·15-s + 0.250·16-s − 0.726·17-s − 0.235i·18-s + 0.304·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.858 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.838248378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.838248378\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-3.29 - 0.383i)T \) |
| good | 5 | \( 1 + 1.58iT - 5T^{2} \) |
| 13 | \( 1 - 2.26T + 13T^{2} \) |
| 17 | \( 1 + 2.99T + 17T^{2} \) |
| 19 | \( 1 - 1.32T + 19T^{2} \) |
| 23 | \( 1 - 2.32T + 23T^{2} \) |
| 29 | \( 1 - 2.90iT - 29T^{2} \) |
| 31 | \( 1 + 10.1iT - 31T^{2} \) |
| 37 | \( 1 + 2.83T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 - 12.0iT - 43T^{2} \) |
| 47 | \( 1 - 0.996iT - 47T^{2} \) |
| 53 | \( 1 + 0.531T + 53T^{2} \) |
| 59 | \( 1 + 0.121iT - 59T^{2} \) |
| 61 | \( 1 + 7.54T + 61T^{2} \) |
| 67 | \( 1 - 0.628T + 67T^{2} \) |
| 71 | \( 1 + 7.26T + 71T^{2} \) |
| 73 | \( 1 - 9.21T + 73T^{2} \) |
| 79 | \( 1 + 11.7iT - 79T^{2} \) |
| 83 | \( 1 - 7.48T + 83T^{2} \) |
| 89 | \( 1 + 11.3iT - 89T^{2} \) |
| 97 | \( 1 - 1.49iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.561641351302733325282224990315, −7.79479511699119944792070072594, −7.10671296383033934505913530974, −6.29311266358705495332904435771, −5.83373353855477530951289580973, −4.73456302041774224365966898272, −4.17302529676902841928419341488, −3.02667297945115463510409813144, −1.65524792623334234264746456388, −0.71861322056479745595595774166,
1.03479094810780211229275401358, 2.28273963483260155341836715335, 3.24365428434162611219478813672, 3.83077517115935319488292321348, 4.67133576235806709934186558997, 5.57749993573253059558552219802, 6.51072491917796517953422429349, 7.09214264280380760670363033917, 8.241893709131364630838422431004, 9.024915493443435684240954733933
