| L(s) = 1 | − 9.72i·3-s − 25i·5-s + 65.1·7-s + 148.·9-s + 71.5i·11-s + 733. i·13-s − 243.·15-s − 716.·17-s + 2.33e3i·19-s − 633. i·21-s − 278.·23-s − 625·25-s − 3.80e3i·27-s + 2.95e3i·29-s − 6.04e3·31-s + ⋯ |
| L(s) = 1 | − 0.623i·3-s − 0.447i·5-s + 0.502·7-s + 0.610·9-s + 0.178i·11-s + 1.20i·13-s − 0.279·15-s − 0.600·17-s + 1.48i·19-s − 0.313i·21-s − 0.109·23-s − 0.200·25-s − 1.00i·27-s + 0.652i·29-s − 1.13·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.842657892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.842657892\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 25iT \) |
| good | 3 | \( 1 + 9.72iT - 243T^{2} \) |
| 7 | \( 1 - 65.1T + 1.68e4T^{2} \) |
| 11 | \( 1 - 71.5iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 733. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 716.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.33e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 278.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.95e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 6.04e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.21e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 3.62e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 230. iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.39e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.24e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 7.28e3iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 1.94e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 1.39e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 5.73e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.55e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.30e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.59e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 2.02e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.20e5T + 8.58e9T^{2} \) |
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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
−11.02172839761618987488366793860, −9.924606699954945606648123792728, −8.971247731666202202748333984661, −7.989698485705439041842817682477, −7.12023362328182720119354219812, −6.15650044497481256161781527330, −4.83286297541393707061905561011, −3.90316657463457265188861634287, −2.01514737070353161492225991134, −1.27768887615432715781919701538,
0.50275881414376860028670259820, 2.20970181028919976607459147024, 3.51251526353820729014078489872, 4.60598454269606263978082954339, 5.58034533650174983424526081802, 6.90345387716106616167022264325, 7.77659650290398804869257173251, 8.934980567824998230980315243593, 9.810011480661996627104578116276, 10.83671720511157391392977170516
