L(s) = 1 | − 2-s − 1.73i·3-s + 4-s + (1.5 + 0.866i)5-s + 1.73i·6-s + (−2 + 1.73i)7-s − 8-s − 2.99·9-s + (−1.5 − 0.866i)10-s + (1.5 − 2.59i)11-s − 1.73i·12-s + (−1 − 3.46i)13-s + (2 − 1.73i)14-s + (1.49 − 2.59i)15-s + 16-s + 6·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.999i·3-s + 0.5·4-s + (0.670 + 0.387i)5-s + 0.707i·6-s + (−0.755 + 0.654i)7-s − 0.353·8-s − 0.999·9-s + (−0.474 − 0.273i)10-s + (0.452 − 0.783i)11-s − 0.499i·12-s + (−0.277 − 0.960i)13-s + (0.534 − 0.462i)14-s + (0.387 − 0.670i)15-s + 0.250·16-s + 1.45·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.600694 - 0.728719i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.600694 - 0.728719i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 + (2 - 1.73i)T \) |
| 13 | \( 1 + (1 + 3.46i)T \) |
good | 5 | \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 3.46iT - 23T^{2} \) |
| 29 | \( 1 + (-7.5 + 4.33i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + (4.5 - 2.59i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.5 + 0.866i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.5 + 0.866i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 10.3iT - 59T^{2} \) |
| 61 | \( 1 + (-1.5 + 0.866i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.5 + 0.866i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.5 - 7.79i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.5 - 11.2i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3.46iT - 83T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 + (9.5 - 16.4i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.39359045466788249225580203444, −9.690985222429625617996817947794, −8.619731516927761445425842223702, −8.068651083609685334264639554292, −6.74699861389655051675361983466, −6.30414506637898754391352688438, −5.41546509918766883852824224614, −3.09614507219141966408188182846, −2.42481906824766069981659866138, −0.69976527940246971659344596072,
1.62082308577913917822821816140, 3.30249027897215130620269749886, 4.32519276222894217755858515757, 5.56908340711862789279267387091, 6.49388866490464604955116058970, 7.56420399145350531179654223740, 8.722501531574962700068674688707, 9.541157999793297379153966001671, 9.999514572158079832985567225461, 10.54747557438362544908538965807