L(s) = 1 | + (−1.52 + 1.29i)2-s + (0.637 − 3.94i)4-s + (−4.27 + 2.59i)5-s − 0.837·7-s + (4.14 + 6.83i)8-s + (3.14 − 9.49i)10-s − 15.7i·11-s + 5.18i·13-s + (1.27 − 1.08i)14-s + (−15.1 − 5.03i)16-s − 27.3i·17-s − 17.9i·19-s + (7.51 + 18.5i)20-s + (20.4 + 24.0i)22-s − 19.1·23-s + ⋯ |
L(s) = 1 | + (−0.761 + 0.648i)2-s + (0.159 − 0.987i)4-s + (−0.854 + 0.518i)5-s − 0.119·7-s + (0.518 + 0.854i)8-s + (0.314 − 0.949i)10-s − 1.43i·11-s + 0.398i·13-s + (0.0910 − 0.0775i)14-s + (−0.949 − 0.314i)16-s − 1.60i·17-s − 0.945i·19-s + (0.375 + 0.926i)20-s + (0.930 + 1.09i)22-s − 0.830·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.375 + 0.926i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.375 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.453367 - 0.305385i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.453367 - 0.305385i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.52 - 1.29i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (4.27 - 2.59i)T \) |
good | 7 | \( 1 + 0.837T + 49T^{2} \) |
| 11 | \( 1 + 15.7iT - 121T^{2} \) |
| 13 | \( 1 - 5.18iT - 169T^{2} \) |
| 17 | \( 1 + 27.3iT - 289T^{2} \) |
| 19 | \( 1 + 17.9iT - 361T^{2} \) |
| 23 | \( 1 + 19.1T + 529T^{2} \) |
| 29 | \( 1 - 45.6T + 841T^{2} \) |
| 31 | \( 1 + 13.6iT - 961T^{2} \) |
| 37 | \( 1 + 15.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 13.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + 27.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 55.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 15.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 87.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 38T + 3.72e3T^{2} \) |
| 67 | \( 1 + 92.2T + 4.48e3T^{2} \) |
| 71 | \( 1 + 130. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 54.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 13.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 59.0T + 6.88e3T^{2} \) |
| 89 | \( 1 + 39.8T + 7.92e3T^{2} \) |
| 97 | \( 1 + 168. iT - 9.40e3T^{2} \) |
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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
−11.75974002972587632914458407893, −11.26180366526703010891687004908, −10.17761196703084499453174804935, −9.008557494089624114951900328337, −8.142295353475451066893814765179, −7.11353738853545099074967595616, −6.21360566618682679034174494619, −4.73935761620860042045907927933, −2.94040598844136140988719943736, −0.42620846727842713117611088831,
1.65352110024393480995398121378, 3.54198900687378576883736996539, 4.63701392677076114813841509614, 6.62595180242563030025377909613, 7.927007482973942068968119387986, 8.411428710941707513684679802200, 9.825942699706752650421159141068, 10.43847137038467360595916284407, 11.75400515311497532125539705876, 12.42132205454013419668267301614