| L(s) = 1 | − i·3-s − 3.23i·7-s − 9-s − 4·11-s − 4.47i·13-s − 0.145i·17-s + 6.38·19-s − 3.23·21-s + 4.61i·23-s + i·27-s − 5.70·29-s − 9.09·31-s + 4i·33-s + 4.76i·37-s − 4.47·39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.22i·7-s − 0.333·9-s − 1.20·11-s − 1.24i·13-s − 0.0353i·17-s + 1.46·19-s − 0.706·21-s + 0.962i·23-s + 0.192i·27-s − 1.05·29-s − 1.63·31-s + 0.696i·33-s + 0.783i·37-s − 0.716·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7793383694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7793383694\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.23iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 4.47iT - 13T^{2} \) |
| 17 | \( 1 + 0.145iT - 17T^{2} \) |
| 19 | \( 1 - 6.38T + 19T^{2} \) |
| 23 | \( 1 - 4.61iT - 23T^{2} \) |
| 29 | \( 1 + 5.70T + 29T^{2} \) |
| 31 | \( 1 + 9.09T + 31T^{2} \) |
| 37 | \( 1 - 4.76iT - 37T^{2} \) |
| 41 | \( 1 + 8.94T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 9.32iT - 47T^{2} \) |
| 53 | \( 1 - 10.0iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 8.85T + 61T^{2} \) |
| 67 | \( 1 + 1.23iT - 67T^{2} \) |
| 71 | \( 1 - 4.47T + 71T^{2} \) |
| 73 | \( 1 + 0.291iT - 73T^{2} \) |
| 79 | \( 1 - 7.32T + 79T^{2} \) |
| 83 | \( 1 + 16.5iT - 83T^{2} \) |
| 89 | \( 1 + 1.70T + 89T^{2} \) |
| 97 | \( 1 - 0.763iT - 97T^{2} \) |
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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
−9.073312940112563011720140683207, −7.87627406669751742601877483316, −7.59764125591555520968693310884, −6.93718358172465514601231246462, −5.54907378710557574318598347845, −5.22760653727360334231224755373, −3.71774914903571449539697336413, −3.01826727702045608959716860047, −1.57123549085851929615064040284, −0.29576730144582873109133683195,
1.96140504071777576560001307853, 2.87828246904884347626846971458, 3.93727990898993317422205845936, 5.16262754315861690654914414626, 5.45991521120101508321690148145, 6.57683041972446562344383350058, 7.57669605921977546480309442442, 8.409012793214130152860153260365, 9.282467111077722216248028050277, 9.589009670510456858071731059846
