| L(s) = 1 | + 5-s + 7-s + 11-s + 3·17-s + 19-s + 3·23-s − 4·25-s + 3·29-s − 2·31-s + 35-s + 5·37-s + 10·41-s + 5·43-s + 8·47-s + 49-s − 6·53-s + 55-s + 6·59-s + 9·61-s − 8·67-s + 2·71-s − 13·73-s + 77-s + 10·79-s − 14·83-s + 3·85-s − 6·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 0.301·11-s + 0.727·17-s + 0.229·19-s + 0.625·23-s − 4/5·25-s + 0.557·29-s − 0.359·31-s + 0.169·35-s + 0.821·37-s + 1.56·41-s + 0.762·43-s + 1.16·47-s + 1/7·49-s − 0.824·53-s + 0.134·55-s + 0.781·59-s + 1.15·61-s − 0.977·67-s + 0.237·71-s − 1.52·73-s + 0.113·77-s + 1.12·79-s − 1.53·83-s + 0.325·85-s − 0.635·89-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
| \(L(1)\) |
\(\approx\) |
\(3.639871074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.639871074\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T + p T^{2} \) |
| 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 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−14.07881759963621, −13.38439054565109, −12.99193807346029, −12.46433795860298, −11.92895941781856, −11.46113792782737, −10.96853390380285, −10.44785897730435, −9.883044765219971, −9.446699319799107, −8.969026947248850, −8.417054781058693, −7.731951693430230, −7.424107453102489, −6.811504073645708, −6.012835845286384, −5.779854589161300, −5.186383479597424, −4.423428591580064, −4.056987289958638, −3.243242727464444, −2.641465256691061, −2.017085142135323, −1.229208812276495, −0.6740732766417821,
0.6740732766417821, 1.229208812276495, 2.017085142135323, 2.641465256691061, 3.243242727464444, 4.056987289958638, 4.423428591580064, 5.186383479597424, 5.779854589161300, 6.012835845286384, 6.811504073645708, 7.424107453102489, 7.731951693430230, 8.417054781058693, 8.969026947248850, 9.446699319799107, 9.883044765219971, 10.44785897730435, 10.96853390380285, 11.46113792782737, 11.92895941781856, 12.46433795860298, 12.99193807346029, 13.38439054565109, 14.07881759963621
