| L(s) = 1 | + (0.174 − 0.119i)2-s + (−1.58 + 1.46i)3-s + (−0.714 + 1.82i)4-s + (3.75 + 1.15i)5-s + (−0.101 + 0.445i)6-s + (0.185 + 0.814i)8-s + (0.124 − 1.66i)9-s + (0.792 − 0.244i)10-s + (−0.0750 − 1.00i)11-s + (−1.54 − 3.93i)12-s + (2.55 + 1.23i)13-s + (−7.63 + 3.67i)15-s + (−2.73 − 2.53i)16-s + (−2.59 − 0.390i)17-s + (−0.176 − 0.305i)18-s + (−1.64 + 2.84i)19-s + ⋯ |
| L(s) = 1 | + (0.123 − 0.0841i)2-s + (−0.914 + 0.848i)3-s + (−0.357 + 0.910i)4-s + (1.67 + 0.517i)5-s + (−0.0414 + 0.181i)6-s + (0.0657 + 0.288i)8-s + (0.0415 − 0.554i)9-s + (0.250 − 0.0772i)10-s + (−0.0226 − 0.301i)11-s + (−0.445 − 1.13i)12-s + (0.708 + 0.341i)13-s + (−1.97 + 0.949i)15-s + (−0.684 − 0.634i)16-s + (−0.628 − 0.0947i)17-s + (−0.0415 − 0.0719i)18-s + (−0.377 + 0.653i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.449 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.635615 + 1.03172i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.635615 + 1.03172i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + (-0.174 + 0.119i)T + (0.730 - 1.86i)T^{2} \) |
| 3 | \( 1 + (1.58 - 1.46i)T + (0.224 - 2.99i)T^{2} \) |
| 5 | \( 1 + (-3.75 - 1.15i)T + (4.13 + 2.81i)T^{2} \) |
| 11 | \( 1 + (0.0750 + 1.00i)T + (-10.8 + 1.63i)T^{2} \) |
| 13 | \( 1 + (-2.55 - 1.23i)T + (8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (2.59 + 0.390i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (1.64 - 2.84i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.67 - 0.403i)T + (21.9 - 6.77i)T^{2} \) |
| 29 | \( 1 + (-2.95 + 3.70i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (-1.33 - 2.31i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.02 - 2.60i)T + (-27.1 + 25.1i)T^{2} \) |
| 41 | \( 1 + (0.845 + 3.70i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (0.320 - 1.40i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (-4.46 + 3.04i)T + (17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (0.112 - 0.286i)T + (-38.8 - 36.0i)T^{2} \) |
| 59 | \( 1 + (-7.68 + 2.37i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (2.16 + 5.52i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (5.38 + 9.33i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.08 - 6.37i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-10.5 - 7.22i)T + (26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + (-2.63 + 4.56i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.55 + 0.748i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (1.00 - 13.4i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 + 7.32T + 97T^{2} \) |
| 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
−11.64026431477945441250991106568, −10.82715111920900537629863640939, −10.08773449142329706178623170105, −9.277768750257775934533307612107, −8.256427967226767937843331886924, −6.65664773007925513978769317645, −5.90629197251665536391046635684, −4.92301651820574870912665764705, −3.77733414191659999880433854947, −2.28817478094062380213628112528,
0.969513588414022192690912008287, 2.06810858409175329577258842865, 4.62718768587296636373165954702, 5.61419225539897529545164176505, 6.14047282448047365548156706573, 6.87591099319861216330339864390, 8.642953068427024518178181543573, 9.449811221306321923006224872104, 10.34901693740873176388689089699, 11.09045996145682922770793419141
