L(s) = 1 | + 2-s − 4-s + 5-s − 2·7-s − 3·8-s + 10-s + 11-s − 2·14-s − 16-s + 2·17-s + 3·19-s − 20-s + 22-s + 25-s + 2·28-s + 5·29-s + 31-s + 5·32-s + 2·34-s − 2·35-s + 5·37-s + 3·38-s − 3·40-s − 8·43-s − 44-s + 2·47-s − 3·49-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.447·5-s − 0.755·7-s − 1.06·8-s + 0.316·10-s + 0.301·11-s − 0.534·14-s − 1/4·16-s + 0.485·17-s + 0.688·19-s − 0.223·20-s + 0.213·22-s + 1/5·25-s + 0.377·28-s + 0.928·29-s + 0.179·31-s + 0.883·32-s + 0.342·34-s − 0.338·35-s + 0.821·37-s + 0.486·38-s − 0.474·40-s − 1.21·43-s − 0.150·44-s + 0.291·47-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.752866740\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.752866740\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−14.17098554568714, −13.54043397478800, −13.25475614388674, −12.84955878849873, −12.20062218715164, −11.81596646469148, −11.41545527514215, −10.41925330543923, −10.01659760732245, −9.746932178314171, −8.996415623858210, −8.729081082094531, −8.050999739765428, −7.309777092349242, −6.767549534097453, −6.136530301504506, −5.792828528405281, −5.193575466640247, −4.607780575314895, −4.058917686803621, −3.315695367589122, −3.019428796646673, −2.253046900695971, −1.227712790433505, −0.5393249746471477,
0.5393249746471477, 1.227712790433505, 2.253046900695971, 3.019428796646673, 3.315695367589122, 4.058917686803621, 4.607780575314895, 5.193575466640247, 5.792828528405281, 6.136530301504506, 6.767549534097453, 7.309777092349242, 8.050999739765428, 8.729081082094531, 8.996415623858210, 9.746932178314171, 10.01659760732245, 10.41925330543923, 11.41545527514215, 11.81596646469148, 12.20062218715164, 12.84955878849873, 13.25475614388674, 13.54043397478800, 14.17098554568714