| L(s) = 1 | + 4·7-s − 3·9-s + 4·11-s − 2·17-s + 4·19-s + 4·23-s − 2·29-s − 8·31-s + 6·37-s + 6·41-s − 8·43-s − 4·47-s + 9·49-s − 6·53-s − 4·59-s − 2·61-s − 12·63-s − 8·67-s − 6·73-s + 16·77-s + 9·81-s + 16·83-s + 6·89-s − 14·97-s − 12·99-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 9-s + 1.20·11-s − 0.485·17-s + 0.917·19-s + 0.834·23-s − 0.371·29-s − 1.43·31-s + 0.986·37-s + 0.937·41-s − 1.21·43-s − 0.583·47-s + 9/7·49-s − 0.824·53-s − 0.520·59-s − 0.256·61-s − 1.51·63-s − 0.977·67-s − 0.702·73-s + 1.82·77-s + 81-s + 1.75·83-s + 0.635·89-s − 1.42·97-s − 1.20·99-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.44447850719243, −14.16166168025802, −13.40465068209489, −13.11512852670146, −12.12024999816672, −11.92269416326696, −11.32863614773167, −11.03231042152344, −10.70452458398871, −9.636677286113645, −9.286618408399620, −8.815346261521540, −8.353498807107048, −7.591172744116420, −7.471658907672824, −6.494720080309416, −6.148315992184757, −5.261643283791490, −5.092199794545160, −4.345133707950524, −3.735337908971623, −3.079139929705046, −2.348069089844300, −1.559732287834174, −1.146218430389943, 0,
1.146218430389943, 1.559732287834174, 2.348069089844300, 3.079139929705046, 3.735337908971623, 4.345133707950524, 5.092199794545160, 5.261643283791490, 6.148315992184757, 6.494720080309416, 7.471658907672824, 7.591172744116420, 8.353498807107048, 8.815346261521540, 9.286618408399620, 9.636677286113645, 10.70452458398871, 11.03231042152344, 11.32863614773167, 11.92269416326696, 12.12024999816672, 13.11512852670146, 13.40465068209489, 14.16166168025802, 14.44447850719243
