L(s) = 1 | + i·2-s − 3-s − 4-s − 0.561i·5-s − i·6-s − i·7-s − i·8-s + 9-s + 0.561·10-s + 1.43i·11-s + 12-s + (−0.561 + 3.56i)13-s + 14-s + 0.561i·15-s + 16-s + 5.68·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.5·4-s − 0.251i·5-s − 0.408i·6-s − 0.377i·7-s − 0.353i·8-s + 0.333·9-s + 0.177·10-s + 0.433i·11-s + 0.288·12-s + (−0.155 + 0.987i)13-s + 0.267·14-s + 0.144i·15-s + 0.250·16-s + 1.37·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.846329 + 0.723343i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.846329 + 0.723343i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 + (0.561 - 3.56i)T \) |
good | 5 | \( 1 + 0.561iT - 5T^{2} \) |
| 11 | \( 1 - 1.43iT - 11T^{2} \) |
| 17 | \( 1 - 5.68T + 17T^{2} \) |
| 19 | \( 1 - 2.56iT - 19T^{2} \) |
| 23 | \( 1 - 5.68T + 23T^{2} \) |
| 29 | \( 1 + 2.56T + 29T^{2} \) |
| 31 | \( 1 - 10.2iT - 31T^{2} \) |
| 37 | \( 1 + 1.68iT - 37T^{2} \) |
| 41 | \( 1 - 4iT - 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 - 6.24iT - 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 12.2iT - 59T^{2} \) |
| 61 | \( 1 - 2.56T + 61T^{2} \) |
| 67 | \( 1 - 7.12iT - 67T^{2} \) |
| 71 | \( 1 + 15.3iT - 71T^{2} \) |
| 73 | \( 1 - 7.43iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 + 8iT - 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−10.91252293023656831853274305266, −10.04087424549474826959752052262, −9.233520644683733315640114958847, −8.223207276587349189906413582017, −7.17424229232181877757388618748, −6.61875524705064131753866320282, −5.36164933279997614873877654856, −4.71893763562569546200998891954, −3.46453253255970694580335035908, −1.31355843075987360880619276384,
0.834707819000932310465833839667, 2.62615273599389016373788112315, 3.63539967962175318920647713713, 5.10285295172301647332546109227, 5.69016212473853088460661458379, 6.97364234987324441591874111064, 8.010271066408051942881125907744, 9.010592278323758335246929041063, 9.989144793989666392904901866589, 10.65293478224083194080861743413