| L(s) = 1 | − i·2-s + i·3-s − 4-s + 2.32i·5-s + 6-s + i·8-s − 9-s + 2.32·10-s + (0.686 + 3.24i)11-s − i·12-s + 1.44·13-s − 2.32·15-s + 16-s − 3.21·17-s + i·18-s − 6.15·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 1.04i·5-s + 0.408·6-s + 0.353i·8-s − 0.333·9-s + 0.735·10-s + (0.207 + 0.978i)11-s − 0.288i·12-s + 0.399·13-s − 0.600·15-s + 0.250·16-s − 0.779·17-s + 0.235i·18-s − 1.41·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.796 - 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.796 - 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9653305077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9653305077\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-0.686 - 3.24i)T \) |
| good | 5 | \( 1 - 2.32iT - 5T^{2} \) |
| 13 | \( 1 - 1.44T + 13T^{2} \) |
| 17 | \( 1 + 3.21T + 17T^{2} \) |
| 19 | \( 1 + 6.15T + 19T^{2} \) |
| 23 | \( 1 - 6.29T + 23T^{2} \) |
| 29 | \( 1 - 3.26iT - 29T^{2} \) |
| 31 | \( 1 - 3.09iT - 31T^{2} \) |
| 37 | \( 1 - 9.14T + 37T^{2} \) |
| 41 | \( 1 + 3.41T + 41T^{2} \) |
| 43 | \( 1 - 1.89iT - 43T^{2} \) |
| 47 | \( 1 - 10.6iT - 47T^{2} \) |
| 53 | \( 1 + 9.21T + 53T^{2} \) |
| 59 | \( 1 + 0.532iT - 59T^{2} \) |
| 61 | \( 1 + 0.467T + 61T^{2} \) |
| 67 | \( 1 + 9.22T + 67T^{2} \) |
| 71 | \( 1 + 3.25T + 71T^{2} \) |
| 73 | \( 1 + 5.00T + 73T^{2} \) |
| 79 | \( 1 + 8.32iT - 79T^{2} \) |
| 83 | \( 1 + 4.56T + 83T^{2} \) |
| 89 | \( 1 - 4.38iT - 89T^{2} \) |
| 97 | \( 1 + 17.8iT - 97T^{2} \) |
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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
−9.084457445001251647031923090744, −8.474269279707110998521534225555, −7.39910203681693703949482584393, −6.68179887981226981848011288790, −5.99769988314923905273768430626, −4.67905990656212284317687130373, −4.38737848104928177893439693104, −3.23051244504975437276886247511, −2.65816304323003449669761783185, −1.56703793285129082202923392115,
0.30931606258439242356411419952, 1.30395931590382363039641120314, 2.60012637817693209405162355325, 3.85316964077575767898260455745, 4.62100733097692476431804813804, 5.42403442655586741750427025464, 6.24914771288926699578081663285, 6.68739012037165343643568172074, 7.72848014150875691128085616887, 8.429631581566400371718328482458
