L(s) = 1 | + (−0.850 − 1.12i)2-s + (−0.553 + 1.92i)4-s + (0.352 − 0.852i)5-s + (−3.43 + 3.43i)7-s + (2.64 − 1.00i)8-s + (−1.26 + 0.325i)10-s + (−1.44 + 3.49i)11-s + (−0.258 + 0.107i)13-s + (6.80 + 0.959i)14-s + (−3.38 − 2.12i)16-s − 5.30·17-s + (2.72 + 6.57i)19-s + (1.44 + 1.14i)20-s + (5.17 − 1.33i)22-s + (2.23 − 2.23i)23-s + ⋯ |
L(s) = 1 | + (−0.601 − 0.798i)2-s + (−0.276 + 0.960i)4-s + (0.157 − 0.381i)5-s + (−1.29 + 1.29i)7-s + (0.934 − 0.356i)8-s + (−0.399 + 0.103i)10-s + (−0.436 + 1.05i)11-s + (−0.0717 + 0.0297i)13-s + (1.81 + 0.256i)14-s + (−0.846 − 0.531i)16-s − 1.28·17-s + (0.625 + 1.50i)19-s + (0.322 + 0.257i)20-s + (1.10 − 0.284i)22-s + (0.466 − 0.466i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.448160 + 0.318647i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.448160 + 0.318647i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.850 + 1.12i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.352 + 0.852i)T + (-3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (3.43 - 3.43i)T - 7iT^{2} \) |
| 11 | \( 1 + (1.44 - 3.49i)T + (-7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (0.258 - 0.107i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + 5.30T + 17T^{2} \) |
| 19 | \( 1 + (-2.72 - 6.57i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-2.23 + 2.23i)T - 23iT^{2} \) |
| 29 | \( 1 + (3.16 - 1.31i)T + (20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + (-1.27 - 0.528i)T + (26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (-5.28 - 5.28i)T + 41iT^{2} \) |
| 43 | \( 1 + (-2.46 - 1.02i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + 0.423iT - 47T^{2} \) |
| 53 | \( 1 + (12.5 + 5.20i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (5.24 + 2.17i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-0.0138 - 0.0333i)T + (-43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (9.82 - 4.06i)T + (47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (-4.64 - 4.64i)T + 71iT^{2} \) |
| 73 | \( 1 + (-3.96 + 3.96i)T - 73iT^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 + (0.867 - 0.359i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-4.82 + 4.82i)T - 89iT^{2} \) |
| 97 | \( 1 - 8.78T + 97T^{2} \) |
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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
−12.11173125989802334038941157698, −11.04265615109192491397909129838, −9.817651310319050645321029245086, −9.412528449332729015132662183481, −8.526735983074984833968427333302, −7.31095770053674154076242544242, −6.06898183961934123261387418224, −4.69677391682791985160631870923, −3.18260348930055807498169491063, −2.05224263218519200625200066443,
0.48366229889020689414483274611, 3.02033639844369138006742898054, 4.55444058218506899332762865486, 6.00832309516361745486060901745, 6.84054405494428307757909040432, 7.48586692681717973483500711212, 8.887276904390247088065365592119, 9.566977121650898320633607261784, 10.72873382603691511888254514512, 10.99985003225069460024967585487