L(s) = 1 | + (0.173 − 0.984i)3-s + (0.907 + 0.761i)5-s + (0.266 + 0.460i)7-s + (−0.939 − 0.342i)9-s + (0.939 − 1.62i)11-s + (−0.673 − 3.82i)13-s + (0.907 − 0.761i)15-s + (−1.09 + 0.397i)17-s + (3.93 + 1.86i)19-s + (0.5 − 0.181i)21-s + (5.13 − 4.30i)23-s + (−0.624 − 3.54i)25-s + (−0.5 + 0.866i)27-s + (3.77 + 1.37i)29-s + (−0.979 − 1.69i)31-s + ⋯ |
L(s) = 1 | + (0.100 − 0.568i)3-s + (0.405 + 0.340i)5-s + (0.100 + 0.174i)7-s + (−0.313 − 0.114i)9-s + (0.283 − 0.490i)11-s + (−0.186 − 1.05i)13-s + (0.234 − 0.196i)15-s + (−0.264 + 0.0964i)17-s + (0.903 + 0.427i)19-s + (0.109 − 0.0397i)21-s + (1.07 − 0.898i)23-s + (−0.124 − 0.708i)25-s + (−0.0962 + 0.166i)27-s + (0.701 + 0.255i)29-s + (−0.175 − 0.304i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.57937 - 0.789187i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.57937 - 0.789187i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.173 + 0.984i)T \) |
| 19 | \( 1 + (-3.93 - 1.86i)T \) |
good | 5 | \( 1 + (-0.907 - 0.761i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.266 - 0.460i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.939 + 1.62i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.673 + 3.82i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (1.09 - 0.397i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-5.13 + 4.30i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-3.77 - 1.37i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (0.979 + 1.69i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 6.88T + 37T^{2} \) |
| 41 | \( 1 + (1.56 - 8.84i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (1.85 + 1.55i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-1.91 - 0.698i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-9.93 + 8.33i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (2.51 - 0.916i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (8.69 - 7.29i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (10.4 + 3.82i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (4.65 + 3.90i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (0.0569 - 0.322i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-2.80 + 15.8i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-5.78 - 10.0i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.618 + 3.50i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (5.52 - 2.01i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.06881685517783487892769755011, −9.028747300951869110528177260027, −8.259185233121225806382106041913, −7.45504500605860259410383067102, −6.47238575690644009746254837944, −5.80666352341782120216757330676, −4.76076498848460182381162928572, −3.28619461679563280672285440963, −2.46462490348901699069921742642, −0.945202638916765697906142328138,
1.42477986384020180366577454544, 2.80011279300455291717957663008, 4.05952900078461909307148297577, 4.85763839093879582831536877961, 5.71121248313364850409824556373, 6.91288697144290212223964113080, 7.58005337583023493074127725198, 9.023420968542684923118905693039, 9.197597557678837991624515590878, 10.06498246314466627123080379137