| 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
−7.51980307623006378307703923640, −7.08936711841097718769845228107, −6.51226426227225953019712640925, −5.74222173379490980600723878384, −4.80855305423902526913920270878, −3.99813678456976911220735431049, −3.57445832842433013958865878941, −2.17282805589372285471373276838, −1.64264331744985843658099004158, −0.44871374973989942399121391387,
0.823435958810173165628360895884, 2.36492736077049564288137584173, 2.92480338692501137479407471075, 3.48743703546211135945917021451, 4.83302498591240462360443360580, 5.34605273724010242482749955261, 5.60019150456867858327299867916, 6.57921374200377357246911826220, 7.64867019098710712962638050258, 7.984350322355258985754390885887
