L(s) = 1 | + (−1.09 + 1.34i)3-s + (2.46 − 1.42i)5-s + (1.02 + 2.44i)7-s + (−0.614 − 2.93i)9-s + (2.42 − 4.20i)11-s + 2.75·13-s + (−0.780 + 4.87i)15-s + (−1.75 + 3.03i)17-s + (−3.14 − 5.44i)19-s + (−4.39 − 1.29i)21-s + (3.15 − 1.82i)23-s + (1.56 − 2.71i)25-s + (4.61 + 2.38i)27-s + 3.90·29-s + (0.858 + 0.495i)31-s + ⋯ |
L(s) = 1 | + (−0.630 + 0.776i)3-s + (1.10 − 0.637i)5-s + (0.385 + 0.922i)7-s + (−0.204 − 0.978i)9-s + (0.731 − 1.26i)11-s + 0.763·13-s + (−0.201 + 1.25i)15-s + (−0.425 + 0.736i)17-s + (−0.721 − 1.24i)19-s + (−0.959 − 0.282i)21-s + (0.658 − 0.380i)23-s + (0.313 − 0.542i)25-s + (0.888 + 0.458i)27-s + 0.725·29-s + (0.154 + 0.0889i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.238i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.59922 + 0.193085i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.59922 + 0.193085i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.09 - 1.34i)T \) |
| 7 | \( 1 + (-1.02 - 2.44i)T \) |
good | 5 | \( 1 + (-2.46 + 1.42i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.42 + 4.20i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2.75T + 13T^{2} \) |
| 17 | \( 1 + (1.75 - 3.03i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.14 + 5.44i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.15 + 1.82i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3.90T + 29T^{2} \) |
| 31 | \( 1 + (-0.858 - 0.495i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.06 + 0.614i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2.10T + 41T^{2} \) |
| 43 | \( 1 - 5.11iT - 43T^{2} \) |
| 47 | \( 1 + (-5.61 - 9.72i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.00 - 1.73i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.890 - 0.514i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.24 - 2.15i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.02 + 2.90i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 9.75iT - 71T^{2} \) |
| 73 | \( 1 + (-0.291 - 0.168i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.80 - 4.85i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 0.138iT - 83T^{2} \) |
| 89 | \( 1 + (0.580 + 1.00i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 11.0iT - 97T^{2} \) |
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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
−10.81609491757944396788211192103, −9.452019443024099240630797576696, −8.946435526663240816551619918886, −8.452348085568079853240481462890, −6.37178540977287307563636255565, −6.09036166589536450998168106612, −5.15359319566322050535920167842, −4.25483869003077469244109569705, −2.80593717430398578178966210624, −1.18373854664409466979465375068,
1.34477211286664604533231132912, 2.27496749400272544180108787600, 4.02207745046063650815403115243, 5.15185893522737674821687007350, 6.24539126802008088393045265743, 6.82335431144988845471950835801, 7.52855203974511712058677032585, 8.711853319147636049179021986247, 9.947844467730981768246264147175, 10.42295093250994558505470524328