| L(s) = 1 | + (−0.0798 − 1.06i)2-s + (0.849 − 0.127i)4-s + (1.57 − 1.45i)5-s + (1.34 + 2.27i)7-s + (−0.679 − 2.97i)8-s + (−1.68 − 1.55i)10-s + (0.197 − 0.134i)11-s + (2.23 + 1.07i)13-s + (2.32 − 1.61i)14-s + (−1.47 + 0.455i)16-s + (1.04 + 2.65i)17-s + (−2.69 − 4.66i)19-s + (1.14 − 1.43i)20-s + (−0.159 − 0.199i)22-s + (−0.535 + 1.36i)23-s + ⋯ |
| L(s) = 1 | + (−0.0564 − 0.753i)2-s + (0.424 − 0.0639i)4-s + (0.703 − 0.652i)5-s + (0.507 + 0.861i)7-s + (−0.240 − 1.05i)8-s + (−0.531 − 0.492i)10-s + (0.0595 − 0.0406i)11-s + (0.618 + 0.298i)13-s + (0.620 − 0.431i)14-s + (−0.369 + 0.113i)16-s + (0.252 + 0.642i)17-s + (−0.617 − 1.07i)19-s + (0.256 − 0.321i)20-s + (−0.0339 − 0.0426i)22-s + (−0.111 + 0.284i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.301 + 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.301 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.48154 - 1.08544i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48154 - 1.08544i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (-1.34 - 2.27i)T \) |
| good | 2 | \( 1 + (0.0798 + 1.06i)T + (-1.97 + 0.298i)T^{2} \) |
| 5 | \( 1 + (-1.57 + 1.45i)T + (0.373 - 4.98i)T^{2} \) |
| 11 | \( 1 + (-0.197 + 0.134i)T + (4.01 - 10.2i)T^{2} \) |
| 13 | \( 1 + (-2.23 - 1.07i)T + (8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (-1.04 - 2.65i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (2.69 + 4.66i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.535 - 1.36i)T + (-16.8 - 15.6i)T^{2} \) |
| 29 | \( 1 + (4.78 - 6.00i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (-2.45 + 4.24i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.21 + 0.334i)T + (35.3 + 10.9i)T^{2} \) |
| 41 | \( 1 + (0.935 + 4.10i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (-2.37 + 10.4i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (0.417 + 5.56i)T + (-46.4 + 7.00i)T^{2} \) |
| 53 | \( 1 + (9.63 - 1.45i)T + (50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (-5.57 - 5.17i)T + (4.40 + 58.8i)T^{2} \) |
| 61 | \( 1 + (5.72 + 0.862i)T + (58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (5.03 - 8.72i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-9.17 - 11.5i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (1.06 - 14.1i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (-0.136 - 0.236i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.50 - 2.16i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (3.43 + 2.34i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
| 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
−11.07151169440658401590420900312, −10.15485872864572209497912675740, −9.123956223302948611925194999414, −8.630359580444064055272926684270, −7.17597712881877468428779132511, −6.06123806761045203049904871858, −5.26454750881979144463443966799, −3.82766871039511792197993941467, −2.37186529881555486404665438817, −1.46477908582773297951246252679,
1.81280785579355893838577780849, 3.21062149055321816023447821367, 4.71546549382243713803314276375, 6.04532345478329618481736213832, 6.49457979712302184878888654812, 7.66329843830372292602945422823, 8.154400779198177515338842771530, 9.561162577933870337559532128786, 10.53156011696739854738815835214, 11.06802730026295213998185360703
