| L(s) = 1 | + (0.966 + 0.902i)2-s + (−0.590 − 1.62i)3-s + (−0.0170 − 0.249i)4-s + (−0.242 + 0.467i)5-s + (0.899 − 2.10i)6-s + (3.06 − 1.33i)7-s + (1.87 − 2.30i)8-s + (−2.30 + 1.92i)9-s + (−0.656 + 0.233i)10-s + (−0.959 + 4.61i)11-s + (−0.396 + 0.175i)12-s + (2.03 − 3.33i)13-s + (4.16 + 1.48i)14-s + (0.903 + 0.118i)15-s + (3.40 − 0.467i)16-s + (−5.34 + 1.10i)17-s + ⋯ |
| L(s) = 1 | + (0.683 + 0.638i)2-s + (−0.340 − 0.940i)3-s + (−0.00854 − 0.124i)4-s + (−0.108 + 0.209i)5-s + (0.367 − 0.860i)6-s + (1.15 − 0.503i)7-s + (0.664 − 0.816i)8-s + (−0.767 + 0.640i)9-s + (−0.207 + 0.0737i)10-s + (−0.289 + 1.39i)11-s + (−0.114 + 0.0505i)12-s + (0.563 − 0.926i)13-s + (1.11 + 0.395i)14-s + (0.233 + 0.0305i)15-s + (0.851 − 0.116i)16-s + (−1.29 + 0.269i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.306i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 + 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.41875 - 0.222510i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.41875 - 0.222510i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.590 + 1.62i)T \) |
| 47 | \( 1 + (-1.48 + 6.69i)T \) |
| good | 2 | \( 1 + (-0.966 - 0.902i)T + (0.136 + 1.99i)T^{2} \) |
| 5 | \( 1 + (0.242 - 0.467i)T + (-2.88 - 4.08i)T^{2} \) |
| 7 | \( 1 + (-3.06 + 1.33i)T + (4.77 - 5.11i)T^{2} \) |
| 11 | \( 1 + (0.959 - 4.61i)T + (-10.0 - 4.38i)T^{2} \) |
| 13 | \( 1 + (-2.03 + 3.33i)T + (-5.98 - 11.5i)T^{2} \) |
| 17 | \( 1 + (5.34 - 1.10i)T + (15.5 - 6.77i)T^{2} \) |
| 19 | \( 1 + (6.86 - 3.55i)T + (10.9 - 15.5i)T^{2} \) |
| 23 | \( 1 + (-1.57 - 1.69i)T + (-1.56 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-2.40 + 1.46i)T + (13.3 - 25.7i)T^{2} \) |
| 31 | \( 1 + (-0.803 - 5.84i)T + (-29.8 + 8.36i)T^{2} \) |
| 37 | \( 1 + (2.15 + 6.05i)T + (-28.7 + 23.3i)T^{2} \) |
| 41 | \( 1 + (-0.437 + 0.356i)T + (8.34 - 40.1i)T^{2} \) |
| 43 | \( 1 + (-0.559 + 0.0382i)T + (42.5 - 5.85i)T^{2} \) |
| 53 | \( 1 + (-2.43 - 2.99i)T + (-10.7 + 51.8i)T^{2} \) |
| 59 | \( 1 + (5.75 + 0.393i)T + (58.4 + 8.03i)T^{2} \) |
| 61 | \( 1 + (-3.22 + 9.07i)T + (-47.3 - 38.4i)T^{2} \) |
| 67 | \( 1 + (2.70 - 6.21i)T + (-45.7 - 48.9i)T^{2} \) |
| 71 | \( 1 + (-4.76 + 4.44i)T + (4.84 - 70.8i)T^{2} \) |
| 73 | \( 1 + (-0.0149 - 0.0533i)T + (-62.3 + 37.9i)T^{2} \) |
| 79 | \( 1 + (6.56 - 9.30i)T + (-26.4 - 74.4i)T^{2} \) |
| 83 | \( 1 + (8.10 + 1.68i)T + (76.1 + 33.0i)T^{2} \) |
| 89 | \( 1 + (2.55 + 1.32i)T + (51.3 + 72.7i)T^{2} \) |
| 97 | \( 1 + (-9.50 - 1.30i)T + (93.4 + 26.1i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.10954046672015747873710217869, −12.51900850562695112448662293751, −10.98207854021726091346540008572, −10.47033257626975593652886508015, −8.448845374380738433510948441292, −7.41129948353523402662201135462, −6.63763244256457627211928324114, −5.37830622486620207754926193467, −4.35871248651546824428685238957, −1.75009447448017482691777320681,
2.59878577763033818041806193597, 4.24469587389504783118463486457, 4.84875960501290737210104532844, 6.28531389092148579656870431560, 8.521399127678019618058888179212, 8.721401077604259873590846799259, 10.80729469364348258534111745802, 11.16344676878964152547713588786, 11.88880818659410723124712888736, 13.18694237770401254026656061952
