L(s) = 1 | − 2·5-s + (−2 + 1.73i)7-s + 4·11-s + (−1.5 + 2.59i)13-s + (3.5 − 6.06i)17-s + (2.5 + 4.33i)19-s + 4·23-s − 25-s + (−0.5 − 0.866i)29-s + (−1.5 − 2.59i)31-s + (4 − 3.46i)35-s + (−5.5 − 9.52i)37-s + (−4.5 + 7.79i)41-s + (2.5 + 4.33i)43-s + (−1.5 + 2.59i)47-s + ⋯ |
L(s) = 1 | − 0.894·5-s + (−0.755 + 0.654i)7-s + 1.20·11-s + (−0.416 + 0.720i)13-s + (0.848 − 1.47i)17-s + (0.573 + 0.993i)19-s + 0.834·23-s − 0.200·25-s + (−0.0928 − 0.160i)29-s + (−0.269 − 0.466i)31-s + (0.676 − 0.585i)35-s + (−0.904 − 1.56i)37-s + (−0.702 + 1.21i)41-s + (0.381 + 0.660i)43-s + (−0.218 + 0.378i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9916202824\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9916202824\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2 - 1.73i)T \) |
good | 5 | \( 1 + 2T + 5T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + (1.5 - 2.59i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.5 + 6.06i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5.5 + 9.52i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.5 - 4.33i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.5 - 6.06i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.5 - 2.59i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.5 - 11.2i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + (3.5 - 6.06i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.5 - 7.79i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.5 - 12.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.5 - 14.7i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.181929144747975200575476902291, −8.170237791056763088531843273729, −7.32602762085257622658210871856, −6.86802397872210985318494587672, −5.89985238707219584250452345723, −5.14236891540570462685648822026, −4.06276014963259268542617735246, −3.47723461502739637357382598701, −2.52665080521329772200878832899, −1.12776323559080478535026871174,
0.36559521605923565424899292189, 1.53337384387615373672462624241, 3.27643704619819400499543470219, 3.50432106540420766680754149511, 4.46729936839499916467067204316, 5.40679823602313623260920547324, 6.40581061938038715067589474531, 7.05920867330372511351829035801, 7.63094317241727098709591640360, 8.522349968588605287546419311556