L(s) = 1 | + (0.380 − 0.658i)2-s + (0.710 + 1.23i)4-s + (1.59 − 2.75i)5-s + (−2.56 − 0.658i)7-s + 2.60·8-s + (−1.21 − 2.09i)10-s + (1.11 + 1.93i)11-s + 3.70·13-s + (−1.40 + 1.43i)14-s + (−0.430 + 0.746i)16-s + (−2.80 − 4.85i)17-s + (−2.21 + 3.82i)19-s + 4.52·20-s + 1.70·22-s + (−0.471 + 0.816i)23-s + ⋯ |
L(s) = 1 | + (0.269 − 0.465i)2-s + (0.355 + 0.615i)4-s + (0.711 − 1.23i)5-s + (−0.968 − 0.249i)7-s + 0.920·8-s + (−0.382 − 0.663i)10-s + (0.337 + 0.584i)11-s + 1.02·13-s + (−0.376 + 0.384i)14-s + (−0.107 + 0.186i)16-s + (−0.679 − 1.17i)17-s + (−0.507 + 0.878i)19-s + 1.01·20-s + 0.363·22-s + (−0.0982 + 0.170i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.757 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.43566 - 0.533783i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.43566 - 0.533783i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.56 + 0.658i)T \) |
good | 2 | \( 1 + (-0.380 + 0.658i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.59 + 2.75i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.11 - 1.93i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 3.70T + 13T^{2} \) |
| 17 | \( 1 + (2.80 + 4.85i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.21 - 3.82i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.471 - 0.816i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 + (-2.85 - 4.93i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.56 - 2.70i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 3.98T + 41T^{2} \) |
| 43 | \( 1 + 3.28T + 43T^{2} \) |
| 47 | \( 1 + (0.112 - 0.195i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.33 - 9.23i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.02 + 1.78i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.92 + 5.05i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.71 + 6.42i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 7.26T + 71T^{2} \) |
| 73 | \( 1 + (3.77 + 6.54i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.41 + 5.91i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 8.11T + 83T^{2} \) |
| 89 | \( 1 + (-4.86 + 8.42i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 0.842T + 97T^{2} \) |
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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
−12.55835834086771301774830027719, −11.74523700019565292406525596386, −10.52050271071881375961147794072, −9.452722215915119019435383632608, −8.641703496719825040839786283806, −7.25794101352671652131950199405, −6.11202225945612341545576712914, −4.66878324491480102281295224713, −3.48145875187955992746860293345, −1.75562215354869559880831019856,
2.21574931045802993404890900932, 3.75142663198476890107936911459, 5.78888646380277477436126506371, 6.30855650846792055215385772322, 7.02424117395271113149223290533, 8.728203315906913205183898444683, 9.894964375118158660220916264576, 10.72686523309820910900740614405, 11.36102449775286229676069431917, 13.21127730334819209769950652315