| L(s) = 1 | + 5-s − 7-s + 5·11-s − 3·17-s − 19-s − 3·23-s − 4·25-s + 3·29-s − 35-s + 3·37-s + 7·43-s + 49-s + 2·53-s + 5·55-s − 8·59-s − 5·61-s − 10·67-s − 9·73-s − 5·77-s + 14·79-s + 6·83-s − 3·85-s − 2·89-s − 95-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s + 1.50·11-s − 0.727·17-s − 0.229·19-s − 0.625·23-s − 4/5·25-s + 0.557·29-s − 0.169·35-s + 0.493·37-s + 1.06·43-s + 1/7·49-s + 0.274·53-s + 0.674·55-s − 1.04·59-s − 0.640·61-s − 1.22·67-s − 1.05·73-s − 0.569·77-s + 1.57·79-s + 0.658·83-s − 0.325·85-s − 0.211·89-s − 0.102·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.97823809404780, −13.78112018402096, −13.30876205598224, −12.64619613991002, −12.10061558651035, −11.87231152747124, −11.17723860368284, −10.70484363801379, −10.16625606430799, −9.518174310603957, −9.267791448037310, −8.798239670194359, −8.140780350197050, −7.555538359691080, −6.938051731122519, −6.376033116700741, −6.090465113535143, −5.564671144413736, −4.598008129065459, −4.272716313590836, −3.690660963335670, −2.991830550618262, −2.261189441563067, −1.686346719721365, −0.9656885977101466, 0,
0.9656885977101466, 1.686346719721365, 2.261189441563067, 2.991830550618262, 3.690660963335670, 4.272716313590836, 4.598008129065459, 5.564671144413736, 6.090465113535143, 6.376033116700741, 6.938051731122519, 7.555538359691080, 8.140780350197050, 8.798239670194359, 9.267791448037310, 9.518174310603957, 10.16625606430799, 10.70484363801379, 11.17723860368284, 11.87231152747124, 12.10061558651035, 12.64619613991002, 13.30876205598224, 13.78112018402096, 13.97823809404780
