| L(s) = 1 | − 1.56i·3-s − 2.56i·7-s + 0.561·9-s − 2·11-s + 3.56i·13-s + 1.43i·17-s + 2·19-s − 4·21-s + i·23-s − 5.56i·27-s + 8.12·29-s + 0.123·31-s + 3.12i·33-s + 0.561i·37-s + 5.56·39-s + ⋯ |
| L(s) = 1 | − 0.901i·3-s − 0.968i·7-s + 0.187·9-s − 0.603·11-s + 0.987i·13-s + 0.348i·17-s + 0.458·19-s − 0.872·21-s + 0.208i·23-s − 1.07i·27-s + 1.50·29-s + 0.0221·31-s + 0.543i·33-s + 0.0923i·37-s + 0.890·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.785536052\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.785536052\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - iT \) |
| good | 3 | \( 1 + 1.56iT - 3T^{2} \) |
| 7 | \( 1 + 2.56iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 3.56iT - 13T^{2} \) |
| 17 | \( 1 - 1.43iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 29 | \( 1 - 8.12T + 29T^{2} \) |
| 31 | \( 1 - 0.123T + 31T^{2} \) |
| 37 | \( 1 - 0.561iT - 37T^{2} \) |
| 41 | \( 1 - 4.12T + 41T^{2} \) |
| 43 | \( 1 + 6.24iT - 43T^{2} \) |
| 47 | \( 1 + 8.68iT - 47T^{2} \) |
| 53 | \( 1 + 8.56iT - 53T^{2} \) |
| 59 | \( 1 - 1.43T + 59T^{2} \) |
| 61 | \( 1 + 0.876T + 61T^{2} \) |
| 67 | \( 1 + 7.43iT - 67T^{2} \) |
| 71 | \( 1 + 5T + 71T^{2} \) |
| 73 | \( 1 + 7.56iT - 73T^{2} \) |
| 79 | \( 1 - 0.876T + 79T^{2} \) |
| 83 | \( 1 + 7.68iT - 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 - 5.12iT - 97T^{2} \) |
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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
−7.927991302606005881101298207196, −7.27500026632007024434469531865, −6.81255675412723937989825555674, −6.14355959632683999539451340381, −5.07968298213947196156874154320, −4.32340726710853324088377595208, −3.53909129052607269127538042456, −2.38219157440588038329753007582, −1.53295005737256853242727538126, −0.55996791465514395342327538120,
1.10570844867588532389902393975, 2.62897703416786599333673922954, 3.01584720279808378849481351837, 4.18421908844144558034966174478, 4.86600892932828006148249790545, 5.51062878384414487217997523856, 6.15721759188241952176603918579, 7.21165241617029307827299166557, 7.926953049957362185075880086013, 8.632670767312308452085475793302
