| L(s) = 1 | − 1.64·5-s − 4.65·7-s − 3.56·11-s − 0.554·13-s − 5.21·17-s − 6.20·19-s + 23-s − 2.29·25-s + 4.55·29-s − 4.55·31-s + 7.66·35-s + 3.54·37-s + 7.84·41-s + 9.33·43-s + 11.3·47-s + 14.6·49-s − 3.66·53-s + 5.86·55-s − 11.1·59-s − 1.56·61-s + 0.912·65-s − 8.38·67-s − 3.84·71-s − 8.97·73-s + 16.6·77-s − 9.76·79-s + 0.434·83-s + ⋯ |
| L(s) = 1 | − 0.736·5-s − 1.76·7-s − 1.07·11-s − 0.153·13-s − 1.26·17-s − 1.42·19-s + 0.208·23-s − 0.458·25-s + 0.845·29-s − 0.818·31-s + 1.29·35-s + 0.582·37-s + 1.22·41-s + 1.42·43-s + 1.65·47-s + 2.09·49-s − 0.503·53-s + 0.791·55-s − 1.45·59-s − 0.200·61-s + 0.113·65-s − 1.02·67-s − 0.456·71-s − 1.05·73-s + 1.89·77-s − 1.09·79-s + 0.0476·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4232318746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4232318746\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 1.64T + 5T^{2} \) |
| 7 | \( 1 + 4.65T + 7T^{2} \) |
| 11 | \( 1 + 3.56T + 11T^{2} \) |
| 13 | \( 1 + 0.554T + 13T^{2} \) |
| 17 | \( 1 + 5.21T + 17T^{2} \) |
| 19 | \( 1 + 6.20T + 19T^{2} \) |
| 29 | \( 1 - 4.55T + 29T^{2} \) |
| 31 | \( 1 + 4.55T + 31T^{2} \) |
| 37 | \( 1 - 3.54T + 37T^{2} \) |
| 41 | \( 1 - 7.84T + 41T^{2} \) |
| 43 | \( 1 - 9.33T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 3.66T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 + 1.56T + 61T^{2} \) |
| 67 | \( 1 + 8.38T + 67T^{2} \) |
| 71 | \( 1 + 3.84T + 71T^{2} \) |
| 73 | \( 1 + 8.97T + 73T^{2} \) |
| 79 | \( 1 + 9.76T + 79T^{2} \) |
| 83 | \( 1 - 0.434T + 83T^{2} \) |
| 89 | \( 1 - 7.02T + 89T^{2} \) |
| 97 | \( 1 - 4.02T + 97T^{2} \) |
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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
−8.851083345425172667655821260988, −7.71245003349360714555129354758, −7.25164594787969580750005734105, −6.28340909166016519968420816454, −5.89209135323419842232442809128, −4.53653191890917402206186955000, −4.02611128600815949456764041967, −2.95640677305405847776877680543, −2.35556987045244911599982836097, −0.35710540578101969505754034865,
0.35710540578101969505754034865, 2.35556987045244911599982836097, 2.95640677305405847776877680543, 4.02611128600815949456764041967, 4.53653191890917402206186955000, 5.89209135323419842232442809128, 6.28340909166016519968420816454, 7.25164594787969580750005734105, 7.71245003349360714555129354758, 8.851083345425172667655821260988
