L(s) = 1 | + (1.14 + 0.831i)2-s + (0.618 + 1.90i)4-s − 2.63i·5-s − 7-s + (−0.874 + 2.68i)8-s + (2.19 − 3.01i)10-s − 4.29i·11-s + 0.0480i·13-s + (−1.14 − 0.831i)14-s + (−3.23 + 2.35i)16-s + 3.16·17-s + 0.726i·19-s + (5.01 − 1.62i)20-s + (3.57 − 4.91i)22-s + 5.28·23-s + ⋯ |
L(s) = 1 | + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s − 1.17i·5-s − 0.377·7-s + (−0.309 + 0.951i)8-s + (0.692 − 0.953i)10-s − 1.29i·11-s + 0.0133i·13-s + (−0.305 − 0.222i)14-s + (−0.809 + 0.587i)16-s + 0.766·17-s + 0.166i·19-s + (1.12 − 0.364i)20-s + (0.761 − 1.04i)22-s + 1.10·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.523119935\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.523119935\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.14 - 0.831i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + 2.63iT - 5T^{2} \) |
| 11 | \( 1 + 4.29iT - 11T^{2} \) |
| 13 | \( 1 - 0.0480iT - 13T^{2} \) |
| 17 | \( 1 - 3.16T + 17T^{2} \) |
| 19 | \( 1 - 0.726iT - 19T^{2} \) |
| 23 | \( 1 - 5.28T + 23T^{2} \) |
| 29 | \( 1 + 8.00iT - 29T^{2} \) |
| 31 | \( 1 - 4.90T + 31T^{2} \) |
| 37 | \( 1 + 9.10iT - 37T^{2} \) |
| 41 | \( 1 + 1.45T + 41T^{2} \) |
| 43 | \( 1 + 8.85iT - 43T^{2} \) |
| 47 | \( 1 - 9.23T + 47T^{2} \) |
| 53 | \( 1 + 3.14iT - 53T^{2} \) |
| 59 | \( 1 - 5.90iT - 59T^{2} \) |
| 61 | \( 1 - 2.19iT - 61T^{2} \) |
| 67 | \( 1 - 8.55iT - 67T^{2} \) |
| 71 | \( 1 - 3.86T + 71T^{2} \) |
| 73 | \( 1 + 7.56T + 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 - 9.05iT - 83T^{2} \) |
| 89 | \( 1 + 5.15T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−9.028081354757477263350775605624, −8.629861247493433010194935495872, −7.81462765827771189676596476941, −6.93748490531742083251653071562, −5.79297803493422739178179815702, −5.53733611540421489476694656187, −4.42963089399238939110868170125, −3.64042599167162622639045491229, −2.59359774266555845703089576407, −0.817550158490106614590981484192,
1.44516254901154103557974918986, 2.78341763004687562675405669807, 3.21214360993236265166156885671, 4.43550712360291902599440986556, 5.19997348765965652853916048932, 6.34306462019328755299321487209, 6.88182417069000837801635130974, 7.56102661270979349920070419369, 8.995851615387229348203394085132, 9.903385182464655341669455443308