| L(s) = 1 | + 3·5-s − 29·7-s + 57·11-s + 20·13-s − 72·17-s + 106·19-s − 174·23-s − 116·25-s − 210·29-s − 47·31-s − 87·35-s + 2·37-s − 6·41-s − 218·43-s − 474·47-s + 498·49-s + 81·53-s + 171·55-s − 84·59-s + 56·61-s + 60·65-s + 142·67-s − 360·71-s − 1.15e3·73-s − 1.65e3·77-s + 160·79-s − 735·83-s + ⋯ |
| L(s) = 1 | + 0.268·5-s − 1.56·7-s + 1.56·11-s + 0.426·13-s − 1.02·17-s + 1.27·19-s − 1.57·23-s − 0.927·25-s − 1.34·29-s − 0.272·31-s − 0.420·35-s + 0.00888·37-s − 0.0228·41-s − 0.773·43-s − 1.47·47-s + 1.45·49-s + 0.209·53-s + 0.419·55-s − 0.185·59-s + 0.117·61-s + 0.114·65-s + 0.258·67-s − 0.601·71-s − 1.85·73-s − 2.44·77-s + 0.227·79-s − 0.972·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p^{3} T^{2} \) |
| 7 | \( 1 + 29 T + p^{3} T^{2} \) |
| 11 | \( 1 - 57 T + p^{3} T^{2} \) |
| 13 | \( 1 - 20 T + p^{3} T^{2} \) |
| 17 | \( 1 + 72 T + p^{3} T^{2} \) |
| 19 | \( 1 - 106 T + p^{3} T^{2} \) |
| 23 | \( 1 + 174 T + p^{3} T^{2} \) |
| 29 | \( 1 + 210 T + p^{3} T^{2} \) |
| 31 | \( 1 + 47 T + p^{3} T^{2} \) |
| 37 | \( 1 - 2 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 T + p^{3} T^{2} \) |
| 43 | \( 1 + 218 T + p^{3} T^{2} \) |
| 47 | \( 1 + 474 T + p^{3} T^{2} \) |
| 53 | \( 1 - 81 T + p^{3} T^{2} \) |
| 59 | \( 1 + 84 T + p^{3} T^{2} \) |
| 61 | \( 1 - 56 T + p^{3} T^{2} \) |
| 67 | \( 1 - 142 T + p^{3} T^{2} \) |
| 71 | \( 1 + 360 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1159 T + p^{3} T^{2} \) |
| 79 | \( 1 - 160 T + p^{3} T^{2} \) |
| 83 | \( 1 + 735 T + p^{3} T^{2} \) |
| 89 | \( 1 + 954 T + p^{3} T^{2} \) |
| 97 | \( 1 - 191 T + p^{3} T^{2} \) |
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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
−9.924169723176414802778001381850, −9.564531867259952754832134677344, −8.692736885487829388125671829700, −7.29890332304040282767090882459, −6.40973041795235868255017603449, −5.80506937285583698149378805024, −4.09087708775382888509235897716, −3.32967836832694412024717777184, −1.73006514770121781742544675875, 0,
1.73006514770121781742544675875, 3.32967836832694412024717777184, 4.09087708775382888509235897716, 5.80506937285583698149378805024, 6.40973041795235868255017603449, 7.29890332304040282767090882459, 8.692736885487829388125671829700, 9.564531867259952754832134677344, 9.924169723176414802778001381850
