| L(s) = 1 | + 4.27·5-s + 4.77i·7-s − 3·9-s − 2.15i·11-s + 0.274·17-s + 4.35i·19-s − 8.71i·23-s + 13.2·25-s + 20.4i·35-s − 7.40i·43-s − 12.8·45-s − 9.07i·47-s − 15.8·49-s − 9.19i·55-s − 3.72·61-s + ⋯ |
| L(s) = 1 | + 1.91·5-s + 1.80i·7-s − 9-s − 0.648i·11-s + 0.0666·17-s + 0.999i·19-s − 1.81i·23-s + 2.65·25-s + 3.45i·35-s − 1.12i·43-s − 1.91·45-s − 1.32i·47-s − 2.26·49-s − 1.23i·55-s − 0.476·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.56615 + 0.419650i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.56615 + 0.419650i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 - 4.27T + 5T^{2} \) |
| 7 | \( 1 - 4.77iT - 7T^{2} \) |
| 11 | \( 1 + 2.15iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 0.274T + 17T^{2} \) |
| 23 | \( 1 + 8.71iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 7.40iT - 43T^{2} \) |
| 47 | \( 1 + 9.07iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 3.72T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 16.8T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 8.71iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−11.94330548691931243202373207318, −10.74638981951057758130655403102, −9.852965391168309228840130012851, −8.797388086100443131135659308373, −8.556699312280834535863333336599, −6.41249982792321442785916107499, −5.82303918883936080912489033550, −5.23864496058199156194338466367, −2.87618227793513116235330023633, −2.07992942805097777018462215339,
1.45004389868267803261193242582, 2.99001804236678869678855341607, 4.63770591680090022804727902045, 5.71154380539050079573623383771, 6.70300565244020127222769166315, 7.63395452662359773499110342714, 9.147925008947517240779884238706, 9.776463114352845364033073039456, 10.58705085985956022967149596302, 11.38252049752915655476169007612
