| L(s) = 1 | − 7-s − 3·9-s + 4·11-s + 2·13-s + 6·17-s + 4·19-s − 23-s − 2·29-s − 4·31-s + 2·37-s − 6·41-s − 8·43-s + 8·47-s + 49-s − 14·53-s − 12·59-s + 6·61-s + 3·63-s − 8·67-s + 2·73-s − 4·77-s + 4·79-s + 9·81-s + 4·83-s − 6·89-s − 2·91-s + 14·97-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s + 1.20·11-s + 0.554·13-s + 1.45·17-s + 0.917·19-s − 0.208·23-s − 0.371·29-s − 0.718·31-s + 0.328·37-s − 0.937·41-s − 1.21·43-s + 1.16·47-s + 1/7·49-s − 1.92·53-s − 1.56·59-s + 0.768·61-s + 0.377·63-s − 0.977·67-s + 0.234·73-s − 0.455·77-s + 0.450·79-s + 81-s + 0.439·83-s − 0.635·89-s − 0.209·91-s + 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.217106604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.217106604\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−14.24906827657222, −13.81168965532448, −13.36177181080556, −12.62576761602398, −12.00928640860695, −11.91847777770244, −11.18957820605622, −10.84122726572783, −10.02080917917310, −9.634566836745504, −9.118628382766415, −8.675211275330249, −8.011057472002961, −7.559809960789418, −6.908469108900960, −6.243572411284126, −5.890247558350615, −5.345090052128297, −4.688879817458616, −3.827931850378451, −3.281733290247282, −3.101937467787166, −1.929873445984178, −1.336913442035872, −0.5267096066977553,
0.5267096066977553, 1.336913442035872, 1.929873445984178, 3.101937467787166, 3.281733290247282, 3.827931850378451, 4.688879817458616, 5.345090052128297, 5.890247558350615, 6.243572411284126, 6.908469108900960, 7.559809960789418, 8.011057472002961, 8.675211275330249, 9.118628382766415, 9.634566836745504, 10.02080917917310, 10.84122726572783, 11.18957820605622, 11.91847777770244, 12.00928640860695, 12.62576761602398, 13.36177181080556, 13.81168965532448, 14.24906827657222
