L(s) = 1 | − 2.29·3-s + 1.65·5-s − 7-s + 2.28·9-s − 6.22·11-s − 0.634·13-s − 3.80·15-s − 5.02·17-s − 2.11·19-s + 2.29·21-s − 8.89·23-s − 2.26·25-s + 1.63·27-s − 1.13·29-s − 8.27·31-s + 14.3·33-s − 1.65·35-s − 2.19·37-s + 1.45·39-s − 4.93·41-s + 5.27·43-s + 3.78·45-s − 6.68·47-s + 49-s + 11.5·51-s + 6.41·53-s − 10.2·55-s + ⋯ |
L(s) = 1 | − 1.32·3-s + 0.739·5-s − 0.377·7-s + 0.762·9-s − 1.87·11-s − 0.175·13-s − 0.981·15-s − 1.21·17-s − 0.486·19-s + 0.501·21-s − 1.85·23-s − 0.453·25-s + 0.315·27-s − 0.210·29-s − 1.48·31-s + 2.49·33-s − 0.279·35-s − 0.360·37-s + 0.233·39-s − 0.769·41-s + 0.805·43-s + 0.563·45-s − 0.975·47-s + 0.142·49-s + 1.61·51-s + 0.880·53-s − 1.38·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1569794422\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1569794422\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + 2.29T + 3T^{2} \) |
| 5 | \( 1 - 1.65T + 5T^{2} \) |
| 11 | \( 1 + 6.22T + 11T^{2} \) |
| 13 | \( 1 + 0.634T + 13T^{2} \) |
| 17 | \( 1 + 5.02T + 17T^{2} \) |
| 19 | \( 1 + 2.11T + 19T^{2} \) |
| 23 | \( 1 + 8.89T + 23T^{2} \) |
| 29 | \( 1 + 1.13T + 29T^{2} \) |
| 31 | \( 1 + 8.27T + 31T^{2} \) |
| 37 | \( 1 + 2.19T + 37T^{2} \) |
| 41 | \( 1 + 4.93T + 41T^{2} \) |
| 43 | \( 1 - 5.27T + 43T^{2} \) |
| 47 | \( 1 + 6.68T + 47T^{2} \) |
| 53 | \( 1 - 6.41T + 53T^{2} \) |
| 59 | \( 1 - 1.50T + 59T^{2} \) |
| 61 | \( 1 + 7.23T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 4.07T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 - 9.19T + 79T^{2} \) |
| 83 | \( 1 - 3.70T + 83T^{2} \) |
| 89 | \( 1 - 1.60T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−7.82609042965399151923782917607, −7.08108308326637975815891881479, −6.26814456097887934728681183171, −5.85482026120912365939872620802, −5.26349323272589692402670145194, −4.64770314485542725031877295736, −3.66656382290573933156410383197, −2.43144961686806233728600646938, −1.91712421895055386759872029072, −0.19879043470868471409665375588,
0.19879043470868471409665375588, 1.91712421895055386759872029072, 2.43144961686806233728600646938, 3.66656382290573933156410383197, 4.64770314485542725031877295736, 5.26349323272589692402670145194, 5.85482026120912365939872620802, 6.26814456097887934728681183171, 7.08108308326637975815891881479, 7.82609042965399151923782917607