| L(s) = 1 | + (1 − i)3-s + 2·7-s + i·9-s + (−1 + i)11-s + (−1 + i)13-s − 2i·17-s + (3 + 3i)19-s + (2 − 2i)21-s + 6·23-s + (4 + 4i)27-s + (−3 − 3i)29-s + 8·31-s + 2i·33-s + (3 + 3i)37-s + 2i·39-s + ⋯ |
| L(s) = 1 | + (0.577 − 0.577i)3-s + 0.755·7-s + 0.333i·9-s + (−0.301 + 0.301i)11-s + (−0.277 + 0.277i)13-s − 0.485i·17-s + (0.688 + 0.688i)19-s + (0.436 − 0.436i)21-s + 1.25·23-s + (0.769 + 0.769i)27-s + (−0.557 − 0.557i)29-s + 1.43·31-s + 0.348i·33-s + (0.493 + 0.493i)37-s + 0.320i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0708i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0708i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.316973432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.316973432\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (-1 + i)T - 3iT^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + (1 - i)T - 11iT^{2} \) |
| 13 | \( 1 + (1 - i)T - 13iT^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + (-3 - 3i)T + 19iT^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + (3 + 3i)T + 29iT^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + (-3 - 3i)T + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (-5 - 5i)T + 43iT^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + (-5 - 5i)T + 53iT^{2} \) |
| 59 | \( 1 + (3 - 3i)T - 59iT^{2} \) |
| 61 | \( 1 + (9 + 9i)T + 61iT^{2} \) |
| 67 | \( 1 + (5 - 5i)T - 67iT^{2} \) |
| 71 | \( 1 + 10iT - 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (-1 + i)T - 83iT^{2} \) |
| 89 | \( 1 - 4iT - 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−9.297236129656082950219034357114, −8.460311740436829434519280396047, −7.70344864331412648208238040484, −7.34469432582924521459797946217, −6.25569424794157170047928943777, −5.11871327533801936140181988879, −4.55024001762475265967683469123, −3.13812131818758889137264684957, −2.27531674827328281987956498482, −1.23374554061319535138905960430,
1.02201724135173927526339978237, 2.57684714268305765122376662778, 3.35078832512169742578649991236, 4.41800258016212945509248578004, 5.11755661289014142830855737583, 6.11345292999740449245689439992, 7.17609075176525688818437269853, 7.934266578093900937375458335451, 8.761414318824999894348447805031, 9.271574097510806860633261607808
