| L(s) = 1 | + (−2.98 − 7.42i)2-s − 6.08i·3-s + (−46.1 + 44.3i)4-s + (−45.1 + 18.1i)6-s − 422. i·7-s + (466. + 209. i)8-s + 691.·9-s + 1.74e3i·11-s + (269. + 280. i)12-s + 898.·13-s + (−3.13e3 + 1.26e3i)14-s + (163. − 4.09e3i)16-s + 5.54e3·17-s + (−2.06e3 − 5.13e3i)18-s − 8.90e3i·19-s + ⋯ |
| L(s) = 1 | + (−0.373 − 0.927i)2-s − 0.225i·3-s + (−0.721 + 0.692i)4-s + (−0.209 + 0.0842i)6-s − 1.23i·7-s + (0.912 + 0.410i)8-s + 0.949·9-s + 1.31i·11-s + (0.156 + 0.162i)12-s + 0.409·13-s + (−1.14 + 0.459i)14-s + (0.0398 − 0.999i)16-s + 1.12·17-s + (−0.354 − 0.880i)18-s − 1.29i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.721 + 0.692i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.721 + 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.539746 - 1.34067i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.539746 - 1.34067i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.98 + 7.42i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 6.08iT - 729T^{2} \) |
| 7 | \( 1 + 422. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 1.74e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 898.T + 4.82e6T^{2} \) |
| 17 | \( 1 - 5.54e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 8.90e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 3.74e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 3.11e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 2.26e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 1.49e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 6.59e3T + 4.75e9T^{2} \) |
| 43 | \( 1 + 8.10e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 3.02e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.72e4T + 2.21e10T^{2} \) |
| 59 | \( 1 + 2.94e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 2.11e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 4.52e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 2.46e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 4.76e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 6.80e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 1.03e6iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 9.37e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + 9.57e5T + 8.32e11T^{2} \) |
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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
−12.37275643050167771241266452189, −11.08313497807820189169926566261, −10.16623185938856439324541808506, −9.402050077116712291224111141501, −7.73279271657646543373339562390, −7.06478335407099073928334593030, −4.73082412849796526622141384543, −3.71944636281861495534651817989, −1.87856577276333321216125698389, −0.65162722073464092104035021661,
1.33752778500833015433176145609, 3.63695089207543363230535656540, 5.37725698665353499684401777084, 6.10842288276604483524722935501, 7.65968733379790116522850414125, 8.646401007077262120149789916746, 9.602785593926512381478190869591, 10.68994648823608102596016107086, 12.12904394715963362491481573115, 13.27848115823032617267448062664
