| L(s) = 1 | + i·3-s − 2.11i·7-s − 9-s + 6.34·11-s − 1.04i·13-s + 1.99i·17-s − 4.30·19-s + 2.11·21-s + 3.11i·23-s − i·27-s + 6.04·29-s + 2.31·31-s + 6.34i·33-s − 4.50i·37-s + 1.04·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.800i·7-s − 0.333·9-s + 1.91·11-s − 0.289i·13-s + 0.484i·17-s − 0.987·19-s + 0.461·21-s + 0.649i·23-s − 0.192i·27-s + 1.12·29-s + 0.415·31-s + 1.10i·33-s − 0.740i·37-s + 0.167·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.016012555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.016012555\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.11iT - 7T^{2} \) |
| 11 | \( 1 - 6.34T + 11T^{2} \) |
| 13 | \( 1 + 1.04iT - 13T^{2} \) |
| 17 | \( 1 - 1.99iT - 17T^{2} \) |
| 19 | \( 1 + 4.30T + 19T^{2} \) |
| 23 | \( 1 - 3.11iT - 23T^{2} \) |
| 29 | \( 1 - 6.04T + 29T^{2} \) |
| 31 | \( 1 - 2.31T + 31T^{2} \) |
| 37 | \( 1 + 4.50iT - 37T^{2} \) |
| 41 | \( 1 + 4.72T + 41T^{2} \) |
| 43 | \( 1 + 5.99iT - 43T^{2} \) |
| 47 | \( 1 + 4.65iT - 47T^{2} \) |
| 53 | \( 1 - 2.18iT - 53T^{2} \) |
| 59 | \( 1 - 1.23T + 59T^{2} \) |
| 61 | \( 1 - 7.82T + 61T^{2} \) |
| 67 | \( 1 - 4.73iT - 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 8.13iT - 73T^{2} \) |
| 79 | \( 1 + 0.884T + 79T^{2} \) |
| 83 | \( 1 + 17.8iT - 83T^{2} \) |
| 89 | \( 1 + 3.20T + 89T^{2} \) |
| 97 | \( 1 + 0.121iT - 97T^{2} \) |
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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
−8.802496898631776807358496500755, −8.141724916171294862580495243527, −7.05792391496506391317827944440, −6.54138084896011680982993254426, −5.72029553394510505576616627383, −4.62798256524760602788566300968, −3.96776237163293697360648774862, −3.43833289570321068655137485859, −1.99722282176986942707704464834, −0.824664661872049742914068254877,
0.989330444651766421091130677308, 2.02103102138184559687378439308, 2.93800482579191985459325675911, 4.05153713250056647166725945839, 4.81166425175273716246095206350, 5.91882142550271483725089091179, 6.60970931385873648860700686319, 6.89347252033542031156523625372, 8.261648502665327355307671352675, 8.570963934408296915378806659368
