L(s) = 1 | + 2-s + 2.28·3-s + 4-s + 5-s + 2.28·6-s − 3.46·7-s + 8-s + 2.22·9-s + 10-s − 3.41·11-s + 2.28·12-s − 2.63·13-s − 3.46·14-s + 2.28·15-s + 16-s − 3.45·17-s + 2.22·18-s + 1.07·19-s + 20-s − 7.93·21-s − 3.41·22-s − 6.75·23-s + 2.28·24-s + 25-s − 2.63·26-s − 1.76·27-s − 3.46·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.32·3-s + 0.5·4-s + 0.447·5-s + 0.933·6-s − 1.31·7-s + 0.353·8-s + 0.743·9-s + 0.316·10-s − 1.03·11-s + 0.660·12-s − 0.731·13-s − 0.926·14-s + 0.590·15-s + 0.250·16-s − 0.839·17-s + 0.525·18-s + 0.246·19-s + 0.223·20-s − 1.73·21-s − 0.728·22-s − 1.40·23-s + 0.466·24-s + 0.200·25-s − 0.517·26-s − 0.338·27-s − 0.655·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 601 | \( 1 + T \) |
good | 3 | \( 1 - 2.28T + 3T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 + 3.41T + 11T^{2} \) |
| 13 | \( 1 + 2.63T + 13T^{2} \) |
| 17 | \( 1 + 3.45T + 17T^{2} \) |
| 19 | \( 1 - 1.07T + 19T^{2} \) |
| 23 | \( 1 + 6.75T + 23T^{2} \) |
| 29 | \( 1 + 1.95T + 29T^{2} \) |
| 31 | \( 1 - 4.65T + 31T^{2} \) |
| 37 | \( 1 + 6.85T + 37T^{2} \) |
| 41 | \( 1 - 0.892T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 2.60T + 47T^{2} \) |
| 53 | \( 1 + 14.2T + 53T^{2} \) |
| 59 | \( 1 + 9.44T + 59T^{2} \) |
| 61 | \( 1 - 9.42T + 61T^{2} \) |
| 67 | \( 1 - 9.09T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 - 3.96T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 0.456T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−7.67062652150338943877844611368, −7.07374060030494468590581495929, −6.23526711869542201110190693254, −5.65525317435326624189783764453, −4.67933015326545087802850279747, −3.88757901100871897122039301836, −3.05825527955010011722365052422, −2.60054270983711483376143536961, −1.92201665217933396262740860895, 0,
1.92201665217933396262740860895, 2.60054270983711483376143536961, 3.05825527955010011722365052422, 3.88757901100871897122039301836, 4.67933015326545087802850279747, 5.65525317435326624189783764453, 6.23526711869542201110190693254, 7.07374060030494468590581495929, 7.67062652150338943877844611368