| L(s) = 1 | + 4·7-s − 13-s − 3·19-s + 4·23-s + 29-s − 8·31-s − 37-s − 41-s + 6·43-s + 11·47-s + 9·49-s + 3·53-s − 10·59-s + 4·61-s + 13·67-s − 9·71-s − 3·79-s − 2·83-s − 10·89-s − 4·91-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 0.277·13-s − 0.688·19-s + 0.834·23-s + 0.185·29-s − 1.43·31-s − 0.164·37-s − 0.156·41-s + 0.914·43-s + 1.60·47-s + 9/7·49-s + 0.412·53-s − 1.30·59-s + 0.512·61-s + 1.58·67-s − 1.06·71-s − 0.337·79-s − 0.219·83-s − 1.05·89-s − 0.419·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.21429556904765, −13.67818303576077, −13.03916913898375, −12.56212597671413, −12.14197576545062, −11.51330542680510, −11.09438389335443, −10.71328070597799, −10.31260407650140, −9.463882852047411, −9.058912681385756, −8.531386174976323, −8.121766034757951, −7.373006157502507, −7.262280919114867, −6.461901774271825, −5.750230304932193, −5.278310114801155, −4.842929849150733, −4.156951627621938, −3.807073900204865, −2.782902924395052, −2.315331442822876, −1.594053064734746, −1.042092719147737, 0,
1.042092719147737, 1.594053064734746, 2.315331442822876, 2.782902924395052, 3.807073900204865, 4.156951627621938, 4.842929849150733, 5.278310114801155, 5.750230304932193, 6.461901774271825, 7.262280919114867, 7.373006157502507, 8.121766034757951, 8.531386174976323, 9.058912681385756, 9.463882852047411, 10.31260407650140, 10.71328070597799, 11.09438389335443, 11.51330542680510, 12.14197576545062, 12.56212597671413, 13.03916913898375, 13.67818303576077, 14.21429556904765
