L(s) = 1 | − 3.22·2-s − 8.71·3-s + 2.37·4-s + 13.5·5-s + 28.0·6-s − 24.0·7-s + 18.1·8-s + 49.0·9-s − 43.5·10-s − 71.6·11-s − 20.6·12-s + 26.7·13-s + 77.5·14-s − 117.·15-s − 77.3·16-s + 35.6·17-s − 157.·18-s + 88.4·19-s + 32.0·20-s + 209.·21-s + 230.·22-s − 64.6·23-s − 158.·24-s + 57.6·25-s − 86.2·26-s − 192.·27-s − 57.1·28-s + ⋯ |
L(s) = 1 | − 1.13·2-s − 1.67·3-s + 0.296·4-s + 1.20·5-s + 1.91·6-s − 1.29·7-s + 0.800·8-s + 1.81·9-s − 1.37·10-s − 1.96·11-s − 0.497·12-s + 0.571·13-s + 1.47·14-s − 2.02·15-s − 1.20·16-s + 0.508·17-s − 2.06·18-s + 1.06·19-s + 0.358·20-s + 2.18·21-s + 2.23·22-s − 0.586·23-s − 1.34·24-s + 0.461·25-s − 0.650·26-s − 1.36·27-s − 0.385·28-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
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + 3.22T + 8T^{2} \) |
| 3 | \( 1 + 8.71T + 27T^{2} \) |
| 5 | \( 1 - 13.5T + 125T^{2} \) |
| 7 | \( 1 + 24.0T + 343T^{2} \) |
| 11 | \( 1 + 71.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 26.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 35.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 88.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 64.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 106.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 7.27T + 2.97e4T^{2} \) |
| 37 | \( 1 + 195.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 404.T + 6.89e4T^{2} \) |
| 47 | \( 1 - 31.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 34.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 486.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 491.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 102.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 119.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 123.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 34.0T + 4.93e5T^{2} \) |
| 83 | \( 1 + 681.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.56e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 93.1T + 9.12e5T^{2} \) |
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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
−8.635751233199801262911262442719, −7.63533921952729762012391382296, −6.85271568399155428018827886710, −6.05931645094097434436167655778, −5.46745960550115079133874997074, −4.84522169772652181644567435615, −3.24753298602404359020814448209, −1.92875544201684334978557938393, −0.792991629104931123358059368856, 0,
0.792991629104931123358059368856, 1.92875544201684334978557938393, 3.24753298602404359020814448209, 4.84522169772652181644567435615, 5.46745960550115079133874997074, 6.05931645094097434436167655778, 6.85271568399155428018827886710, 7.63533921952729762012391382296, 8.635751233199801262911262442719