| L(s) = 1 | − 2·3-s + 9-s + 11-s + 2·13-s − 4·17-s − 5·23-s + 4·27-s − 3·29-s − 10·31-s − 2·33-s + 5·37-s − 4·39-s + 10·41-s + 5·43-s − 4·47-s + 8·51-s + 10·53-s − 10·59-s − 10·61-s − 5·67-s + 10·69-s − 3·71-s + 10·73-s − 13·79-s − 11·81-s − 10·83-s + 6·87-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.970·17-s − 1.04·23-s + 0.769·27-s − 0.557·29-s − 1.79·31-s − 0.348·33-s + 0.821·37-s − 0.640·39-s + 1.56·41-s + 0.762·43-s − 0.583·47-s + 1.12·51-s + 1.37·53-s − 1.30·59-s − 1.28·61-s − 0.610·67-s + 1.20·69-s − 0.356·71-s + 1.17·73-s − 1.46·79-s − 1.22·81-s − 1.09·83-s + 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7722770591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7722770591\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−16.01245484708525, −15.20089763587318, −14.60938391657761, −14.05713085652313, −13.45643226017812, −12.73627911688430, −12.47024990918852, −11.67993740036903, −11.21830522877548, −10.91762757245836, −10.33005070747817, −9.504743876833539, −9.036818540667050, −8.420110036578327, −7.569730659894561, −7.104799502053364, −6.253957793599983, −5.917863224439586, −5.455456641078122, −4.442213986000207, −4.162538909975671, −3.203281058605212, −2.253031813826792, −1.430817160206086, −0.4003222109907981,
0.4003222109907981, 1.430817160206086, 2.253031813826792, 3.203281058605212, 4.162538909975671, 4.442213986000207, 5.455456641078122, 5.917863224439586, 6.253957793599983, 7.104799502053364, 7.569730659894561, 8.420110036578327, 9.036818540667050, 9.504743876833539, 10.33005070747817, 10.91762757245836, 11.21830522877548, 11.67993740036903, 12.47024990918852, 12.73627911688430, 13.45643226017812, 14.05713085652313, 14.60938391657761, 15.20089763587318, 16.01245484708525
