L(s) = 1 | + (−0.5 − 0.866i)5-s + (2.5 − 4.33i)11-s − 3·17-s − 5·19-s + (−3 − 5.19i)23-s + (−0.499 + 0.866i)25-s + (−5 + 8.66i)29-s + (−1 − 1.73i)31-s + 4·37-s + (−1.5 − 2.59i)41-s + (1.5 − 2.59i)43-s + (−2 + 3.46i)47-s + (3.5 + 6.06i)49-s + 6·53-s − 5·55-s + ⋯ |
L(s) = 1 | + (−0.223 − 0.387i)5-s + (0.753 − 1.30i)11-s − 0.727·17-s − 1.14·19-s + (−0.625 − 1.08i)23-s + (−0.0999 + 0.173i)25-s + (−0.928 + 1.60i)29-s + (−0.179 − 0.311i)31-s + 0.657·37-s + (−0.234 − 0.405i)41-s + (0.228 − 0.396i)43-s + (−0.291 + 0.505i)47-s + (0.5 + 0.866i)49-s + 0.824·53-s − 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6436018044\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6436018044\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
good | 7 | \( 1 + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (5 - 8.66i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.5 + 2.59i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2 - 3.46i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 + 15T + 73T^{2} \) |
| 79 | \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + (-6.5 + 11.2i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.940403965776297884536880497636, −8.114780567337784726864762637992, −7.14998900217653976309035900039, −6.28680717671166035417709363644, −5.70733891978350759841606202092, −4.51604435548213052849703621204, −3.91767374355500027577096924479, −2.84072250887378718078314778442, −1.59168597887098548294120467714, −0.21552206557671963973280089429,
1.69561167442075605649894026703, 2.53432958240295813330293143094, 3.98978339108529072480769504268, 4.26989429313363851772515879910, 5.52730855089390466044360912338, 6.42215607730218376178347788465, 7.06717807493524218520804241978, 7.78766142118267506199257065619, 8.655241520453248167470365463302, 9.527254866038541719902160716674