| L(s) = 1 | + 3-s + 5-s − 7-s − 2·9-s + 4·11-s + 15-s − 6·17-s + 4·19-s − 21-s + 23-s − 4·25-s − 5·27-s + 6·29-s − 6·31-s + 4·33-s − 35-s − 4·37-s + 8·41-s + 4·43-s − 2·45-s − 6·47-s + 49-s − 6·51-s − 4·53-s + 4·55-s + 4·57-s + 3·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s − 2/3·9-s + 1.20·11-s + 0.258·15-s − 1.45·17-s + 0.917·19-s − 0.218·21-s + 0.208·23-s − 4/5·25-s − 0.962·27-s + 1.11·29-s − 1.07·31-s + 0.696·33-s − 0.169·35-s − 0.657·37-s + 1.24·41-s + 0.609·43-s − 0.298·45-s − 0.875·47-s + 1/7·49-s − 0.840·51-s − 0.549·53-s + 0.539·55-s + 0.529·57-s + 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.313855808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.313855808\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.06719815667484, −13.70481685568313, −13.14483647809264, −12.69127248676506, −11.87895922100388, −11.72388453118001, −11.01609418918562, −10.62680312560148, −9.816203364296231, −9.377637942080902, −9.036451882357658, −8.685312104084797, −7.954385459714389, −7.392573348443623, −6.804895829290708, −6.270011940120464, −5.846340380695601, −5.218429743007669, −4.386555046387242, −3.997536347836099, −3.178671462782171, −2.835338295415528, −2.000320559178160, −1.502852317583995, −0.4640061992108648,
0.4640061992108648, 1.502852317583995, 2.000320559178160, 2.835338295415528, 3.178671462782171, 3.997536347836099, 4.386555046387242, 5.218429743007669, 5.846340380695601, 6.270011940120464, 6.804895829290708, 7.392573348443623, 7.954385459714389, 8.685312104084797, 9.036451882357658, 9.377637942080902, 9.816203364296231, 10.62680312560148, 11.01609418918562, 11.72388453118001, 11.87895922100388, 12.69127248676506, 13.14483647809264, 13.70481685568313, 14.06719815667484
