| L(s) = 1 | − 3.68·2-s − 8.25·3-s + 5.56·4-s − 12.2·5-s + 30.4·6-s + 28.5·7-s + 8.95·8-s + 41.1·9-s + 45.2·10-s − 55.8·11-s − 45.9·12-s − 70.9·13-s − 105.·14-s + 101.·15-s − 77.5·16-s + 20.9·17-s − 151.·18-s + 25.4·19-s − 68.4·20-s − 235.·21-s + 205.·22-s + 103.·23-s − 73.9·24-s + 26.1·25-s + 261.·26-s − 116.·27-s + 159.·28-s + ⋯ |
| L(s) = 1 | − 1.30·2-s − 1.58·3-s + 0.696·4-s − 1.09·5-s + 2.06·6-s + 1.54·7-s + 0.395·8-s + 1.52·9-s + 1.43·10-s − 1.53·11-s − 1.10·12-s − 1.51·13-s − 2.00·14-s + 1.74·15-s − 1.21·16-s + 0.298·17-s − 1.98·18-s + 0.307·19-s − 0.765·20-s − 2.44·21-s + 1.99·22-s + 0.940·23-s − 0.628·24-s + 0.208·25-s + 1.97·26-s − 0.832·27-s + 1.07·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2046622258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2046622258\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 3.68T + 8T^{2} \) |
| 3 | \( 1 + 8.25T + 27T^{2} \) |
| 5 | \( 1 + 12.2T + 125T^{2} \) |
| 7 | \( 1 - 28.5T + 343T^{2} \) |
| 11 | \( 1 + 55.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 70.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 20.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 25.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 103.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 94.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 16.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 23.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 75.5T + 6.89e4T^{2} \) |
| 47 | \( 1 + 5.56T + 1.03e5T^{2} \) |
| 53 | \( 1 - 622.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 189.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 371.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 620.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 644.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 488.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.11e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 351.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 429.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.18e3T + 9.12e5T^{2} \) |
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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
−8.715072956088944109813178600057, −7.923285662469227792794556432471, −7.53007193108795251406183083102, −6.95535577204364844989213361275, −5.39875170739769868338659081756, −4.97588280697225545195116245796, −4.36812653868735879968532243774, −2.51494222840177498346517213352, −1.22922314295521618478944550999, −0.32469877060187158386489905337,
0.32469877060187158386489905337, 1.22922314295521618478944550999, 2.51494222840177498346517213352, 4.36812653868735879968532243774, 4.97588280697225545195116245796, 5.39875170739769868338659081756, 6.95535577204364844989213361275, 7.53007193108795251406183083102, 7.923285662469227792794556432471, 8.715072956088944109813178600057
