| L(s) = 1 | − 0.0489i·3-s − 3.24i·5-s + 0.683i·7-s + 2.99·9-s − 4.91i·11-s + 3.27·13-s − 0.158·15-s + (−3.23 + 2.55i)17-s − 4.26·19-s + 0.0334·21-s − 5.93i·23-s − 5.50·25-s − 0.293i·27-s − 3.54i·29-s − 1.31i·31-s + ⋯ |
| L(s) = 1 | − 0.0282i·3-s − 1.44i·5-s + 0.258i·7-s + 0.999·9-s − 1.48i·11-s + 0.909·13-s − 0.0409·15-s + (−0.784 + 0.620i)17-s − 0.978·19-s + 0.00730·21-s − 1.23i·23-s − 1.10·25-s − 0.0565i·27-s − 0.658i·29-s − 0.235i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.655679131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.655679131\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (3.23 - 2.55i)T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 0.0489iT - 3T^{2} \) |
| 5 | \( 1 + 3.24iT - 5T^{2} \) |
| 7 | \( 1 - 0.683iT - 7T^{2} \) |
| 11 | \( 1 + 4.91iT - 11T^{2} \) |
| 13 | \( 1 - 3.27T + 13T^{2} \) |
| 19 | \( 1 + 4.26T + 19T^{2} \) |
| 23 | \( 1 + 5.93iT - 23T^{2} \) |
| 29 | \( 1 + 3.54iT - 29T^{2} \) |
| 31 | \( 1 + 1.31iT - 31T^{2} \) |
| 37 | \( 1 + 2.49iT - 37T^{2} \) |
| 41 | \( 1 - 0.139iT - 41T^{2} \) |
| 43 | \( 1 - 0.999T + 43T^{2} \) |
| 47 | \( 1 - 5.28T + 47T^{2} \) |
| 53 | \( 1 + 7.01T + 53T^{2} \) |
| 61 | \( 1 - 2.89iT - 61T^{2} \) |
| 67 | \( 1 + 9.43T + 67T^{2} \) |
| 71 | \( 1 + 5.35iT - 71T^{2} \) |
| 73 | \( 1 - 8.00iT - 73T^{2} \) |
| 79 | \( 1 + 8.36iT - 79T^{2} \) |
| 83 | \( 1 - 8.80T + 83T^{2} \) |
| 89 | \( 1 + 4.01T + 89T^{2} \) |
| 97 | \( 1 - 10.6iT - 97T^{2} \) |
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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
−8.405450677472067982974676357342, −7.64736267243981985794197986892, −6.33994514408013150626695149557, −6.12216140240301843522557544378, −5.09612466425869259870354597962, −4.27113302139572766587920637399, −3.83603048316681392978525836286, −2.42618422278976335901419596772, −1.37624559044308742087421482561, −0.48604636335217700985887294610,
1.51904825145918913351673573828, 2.32967107754003392141408388990, 3.35927169222414829629050708765, 4.14012399803147057603520527257, 4.80399093917687246240808738247, 6.00262792597799397060788599693, 6.78904749311115303296851927594, 7.13274619076692133151417015713, 7.68906682149466844429695514135, 8.809877667262801445527900401938
