| L(s) = 1 | + i·3-s + 3.54i·7-s − 9-s − 2.20·11-s − 7.17i·13-s − 6.36i·17-s + 2.31·19-s − 3.54·21-s + 2.17i·23-s − i·27-s − 0.847·29-s − 4.30·31-s − 2.20i·33-s + 7.22i·37-s + 7.17·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.34i·7-s − 0.333·9-s − 0.665·11-s − 1.99i·13-s − 1.54i·17-s + 0.530·19-s − 0.774·21-s + 0.453i·23-s − 0.192i·27-s − 0.157·29-s − 0.772·31-s − 0.384i·33-s + 1.18i·37-s + 1.14·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.481334677\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.481334677\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 3.54iT - 7T^{2} \) |
| 11 | \( 1 + 2.20T + 11T^{2} \) |
| 13 | \( 1 + 7.17iT - 13T^{2} \) |
| 17 | \( 1 + 6.36iT - 17T^{2} \) |
| 19 | \( 1 - 2.31T + 19T^{2} \) |
| 23 | \( 1 - 2.17iT - 23T^{2} \) |
| 29 | \( 1 + 0.847T + 29T^{2} \) |
| 31 | \( 1 + 4.30T + 31T^{2} \) |
| 37 | \( 1 - 7.22iT - 37T^{2} \) |
| 41 | \( 1 + 1.34T + 41T^{2} \) |
| 43 | \( 1 - 8.18iT - 43T^{2} \) |
| 47 | \( 1 + 6.05iT - 47T^{2} \) |
| 53 | \( 1 - 11.9iT - 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 + 6.92T + 61T^{2} \) |
| 67 | \( 1 + 4.73iT - 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 - 1.08iT - 73T^{2} \) |
| 79 | \( 1 - 5.84T + 79T^{2} \) |
| 83 | \( 1 - 12.5iT - 83T^{2} \) |
| 89 | \( 1 - 7.02T + 89T^{2} \) |
| 97 | \( 1 - 18.1iT - 97T^{2} \) |
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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
−7.984350322355258985754390885887, −7.64867019098710712962638050258, −6.57921374200377357246911826220, −5.60019150456867858327299867916, −5.34605273724010242482749955261, −4.83302498591240462360443360580, −3.48743703546211135945917021451, −2.92480338692501137479407471075, −2.36492736077049564288137584173, −0.823435958810173165628360895884,
0.44871374973989942399121391387, 1.64264331744985843658099004158, 2.17282805589372285471373276838, 3.57445832842433013958865878941, 3.99813678456976911220735431049, 4.80855305423902526913920270878, 5.74222173379490980600723878384, 6.51226426227225953019712640925, 7.08936711841097718769845228107, 7.51980307623006378307703923640
