| L(s) = 1 | + i·2-s − 4-s − 0.115·5-s − i·8-s − 0.115i·10-s + 3.52i·11-s − 2.01i·13-s + 16-s + 2.44·17-s − 5.10i·19-s + 0.115·20-s − 3.52·22-s − 5.73i·23-s − 4.98·25-s + 2.01·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.0517·5-s − 0.353i·8-s − 0.0365i·10-s + 1.06i·11-s − 0.557i·13-s + 0.250·16-s + 0.594·17-s − 1.17i·19-s + 0.0258·20-s − 0.751·22-s − 1.19i·23-s − 0.997·25-s + 0.394·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.284141332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.284141332\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 0.115T + 5T^{2} \) |
| 11 | \( 1 - 3.52iT - 11T^{2} \) |
| 13 | \( 1 + 2.01iT - 13T^{2} \) |
| 17 | \( 1 - 2.44T + 17T^{2} \) |
| 19 | \( 1 + 5.10iT - 19T^{2} \) |
| 23 | \( 1 + 5.73iT - 23T^{2} \) |
| 29 | \( 1 + 4.48iT - 29T^{2} \) |
| 31 | \( 1 + 1.03iT - 31T^{2} \) |
| 37 | \( 1 - 4.69T + 37T^{2} \) |
| 41 | \( 1 + 4.06T + 41T^{2} \) |
| 43 | \( 1 + 6.70T + 43T^{2} \) |
| 47 | \( 1 + 2.96T + 47T^{2} \) |
| 53 | \( 1 - 9.17iT - 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 2.73iT - 61T^{2} \) |
| 67 | \( 1 + 2.54T + 67T^{2} \) |
| 71 | \( 1 + 5.12iT - 71T^{2} \) |
| 73 | \( 1 + 7.60iT - 73T^{2} \) |
| 79 | \( 1 + 4.88T + 79T^{2} \) |
| 83 | \( 1 - 17.9T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 + 12.2iT - 97T^{2} \) |
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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
−8.656256082031946636767256138763, −7.939968133540248723428524212049, −7.32328562759614440879532556288, −6.55820398894183316153696279396, −5.80541323823095249454138541479, −4.86174256875075729053402438600, −4.31248349157510695528623689249, −3.15186325093280909511067557649, −2.04972700747739280179827306414, −0.47156665946052197959466610329,
1.13143247279278337504857166906, 2.10669983750499021186481286866, 3.47122856692315208708039974270, 3.69026142469652912070725290395, 5.01412001608376159297325315632, 5.68933506291116090496447485817, 6.51284422573863413226270461018, 7.61284946269107612091030631535, 8.242745659264991433541617634684, 8.964380954166193761550833082391
