| L(s) = 1 | + 5-s + 5·7-s + 5·11-s + 4·13-s − 6·17-s + 2·19-s − 6·23-s + 5·25-s − 6·29-s + 5·31-s + 5·35-s + 2·37-s + 16·41-s + 8·43-s + 4·47-s + 18·49-s − 9·53-s + 5·55-s − 3·59-s + 12·61-s + 4·65-s + 2·67-s + 16·71-s + 14·73-s + 25·77-s + 79-s − 34·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.88·7-s + 1.50·11-s + 1.10·13-s − 1.45·17-s + 0.458·19-s − 1.25·23-s + 25-s − 1.11·29-s + 0.898·31-s + 0.845·35-s + 0.328·37-s + 2.49·41-s + 1.21·43-s + 0.583·47-s + 18/7·49-s − 1.23·53-s + 0.674·55-s − 0.390·59-s + 1.53·61-s + 0.496·65-s + 0.244·67-s + 1.89·71-s + 1.63·73-s + 2.84·77-s + 0.112·79-s − 3.73·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.773254889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.773254889\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.10051905916083327559476162561, −9.656186245915957256870375452217, −9.328792234842016858583479085048, −8.857126283461275325925549495738, −8.419191577310011041877260697941, −8.396202720501077644362789395296, −7.52725885682715943320558593367, −7.43385166399178688060561871033, −6.75777853345412627581954342757, −6.18510876913338181325759321643, −6.03549181075314301035101084618, −5.48092315071384705466987905613, −4.83810623470018991429148729167, −4.45597551617817180174755871541, −3.90126648109492614393109521578, −3.79488237679497876746623222698, −2.41200872600283332440526837655, −2.35038030609312245103349140896, −1.33905286141008585789474465930, −1.10121837595018112655442827978,
1.10121837595018112655442827978, 1.33905286141008585789474465930, 2.35038030609312245103349140896, 2.41200872600283332440526837655, 3.79488237679497876746623222698, 3.90126648109492614393109521578, 4.45597551617817180174755871541, 4.83810623470018991429148729167, 5.48092315071384705466987905613, 6.03549181075314301035101084618, 6.18510876913338181325759321643, 6.75777853345412627581954342757, 7.43385166399178688060561871033, 7.52725885682715943320558593367, 8.396202720501077644362789395296, 8.419191577310011041877260697941, 8.857126283461275325925549495738, 9.328792234842016858583479085048, 9.656186245915957256870375452217, 10.10051905916083327559476162561
