| L(s) = 1 | + 16·2-s + 256·4-s − 2.41e3i·5-s + 2.48e3·7-s + 4.09e3·8-s − 3.87e4i·10-s + 3.08e4i·11-s − 4.36e4i·13-s + 3.97e4·14-s + 6.55e4·16-s + 1.07e5i·17-s + (−5.33e5 + 1.95e5i)19-s − 6.19e5i·20-s + 4.93e5i·22-s + 2.64e4i·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.73i·5-s + 0.390·7-s + 0.353·8-s − 1.22i·10-s + 0.635i·11-s − 0.424i·13-s + 0.276·14-s + 0.250·16-s + 0.311i·17-s + (−0.938 + 0.344i)19-s − 0.865i·20-s + 0.449i·22-s + 0.0196i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.823 - 0.567i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.6715236900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6715236900\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 16T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (5.33e5 - 1.95e5i)T \) |
| good | 5 | \( 1 + 2.41e3iT - 1.95e6T^{2} \) |
| 7 | \( 1 - 2.48e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.08e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 + 4.36e4iT - 1.06e10T^{2} \) |
| 17 | \( 1 - 1.07e5iT - 1.18e11T^{2} \) |
| 23 | \( 1 - 2.64e4iT - 1.80e12T^{2} \) |
| 29 | \( 1 + 1.30e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.57e6iT - 2.64e13T^{2} \) |
| 37 | \( 1 - 2.85e5iT - 1.29e14T^{2} \) |
| 41 | \( 1 + 3.37e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.81e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.01e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 + 6.20e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.73e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 8.54e6T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.18e8iT - 2.72e16T^{2} \) |
| 71 | \( 1 + 1.41e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.16e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.10e8iT - 1.19e17T^{2} \) |
| 83 | \( 1 + 1.66e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 - 2.17e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 3.97e7iT - 7.60e17T^{2} \) |
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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
−9.362345430375834637357272084470, −8.397051140329102826723015936618, −7.69482737942286594925189412172, −6.28592191091285618416444365993, −5.28194432751999541572076980820, −4.62712125428802939953763638952, −3.77926776627420695883065673702, −2.12149943827887936020167455361, −1.31644341158231349942892306046, −0.084503024498718198787006794904,
1.71726982146918416929621674712, 2.74800357857864072630767034850, 3.48938561946542309574580636495, 4.62852515542017652808301261988, 5.91151280516917671289051398171, 6.67076322561861677067255562075, 7.39188756519513840652394200645, 8.538982714084177103757963549152, 9.897318483771161512559977683887, 10.90212181144087362595351120830
