| L(s) = 1 | + 2-s − 0.0511·3-s + 4-s − 0.949·5-s − 0.0511·6-s + 8-s − 2.99·9-s − 0.949·10-s + 0.0646·11-s − 0.0511·12-s − 3.36·13-s + 0.0485·15-s + 16-s − 4.94·17-s − 2.99·18-s + 0.771·19-s − 0.949·20-s + 0.0646·22-s + 4.13·23-s − 0.0511·24-s − 4.09·25-s − 3.36·26-s + 0.306·27-s + 8.87·29-s + 0.0485·30-s + 9.05·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0295·3-s + 0.5·4-s − 0.424·5-s − 0.0208·6-s + 0.353·8-s − 0.999·9-s − 0.300·10-s + 0.0194·11-s − 0.0147·12-s − 0.932·13-s + 0.0125·15-s + 0.250·16-s − 1.19·17-s − 0.706·18-s + 0.176·19-s − 0.212·20-s + 0.0137·22-s + 0.862·23-s − 0.0104·24-s − 0.819·25-s − 0.659·26-s + 0.0590·27-s + 1.64·29-s + 0.00887·30-s + 1.62·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.257983564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.257983564\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + 0.0511T + 3T^{2} \) |
| 5 | \( 1 + 0.949T + 5T^{2} \) |
| 11 | \( 1 - 0.0646T + 11T^{2} \) |
| 13 | \( 1 + 3.36T + 13T^{2} \) |
| 17 | \( 1 + 4.94T + 17T^{2} \) |
| 19 | \( 1 - 0.771T + 19T^{2} \) |
| 23 | \( 1 - 4.13T + 23T^{2} \) |
| 29 | \( 1 - 8.87T + 29T^{2} \) |
| 31 | \( 1 - 9.05T + 31T^{2} \) |
| 37 | \( 1 - 7.65T + 37T^{2} \) |
| 43 | \( 1 - 6.78T + 43T^{2} \) |
| 47 | \( 1 - 9.14T + 47T^{2} \) |
| 53 | \( 1 - 1.26T + 53T^{2} \) |
| 59 | \( 1 - 2.61T + 59T^{2} \) |
| 61 | \( 1 - 3.21T + 61T^{2} \) |
| 67 | \( 1 - 4.96T + 67T^{2} \) |
| 71 | \( 1 - 5.15T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 + 8.53T + 89T^{2} \) |
| 97 | \( 1 - 8.70T + 97T^{2} \) |
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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
−8.312348802874567761823868083621, −7.70371073782115259331698536819, −6.81367870754752721614593963430, −6.24406100676509439391421529026, −5.36906783203957653823147351530, −4.62635825646065392080208709279, −4.02008953973364509399558075539, −2.76426178808080353756808163033, −2.48855844033234637722661358898, −0.75126019591091201509550359006,
0.75126019591091201509550359006, 2.48855844033234637722661358898, 2.76426178808080353756808163033, 4.02008953973364509399558075539, 4.62635825646065392080208709279, 5.36906783203957653823147351530, 6.24406100676509439391421529026, 6.81367870754752721614593963430, 7.70371073782115259331698536819, 8.312348802874567761823868083621
