| L(s) = 1 | + (0.893 − 0.448i)2-s + (1.51 − 0.845i)3-s + (0.597 − 0.802i)4-s + (−0.379 + 1.26i)5-s + (0.971 − 1.43i)6-s + (−0.768 + 1.78i)7-s + (0.173 − 0.984i)8-s + (1.57 − 2.55i)9-s + (0.229 + 1.30i)10-s + (−2.10 − 0.499i)11-s + (0.224 − 1.71i)12-s + (−1.69 + 1.11i)13-s + (0.112 + 1.93i)14-s + (0.497 + 2.23i)15-s + (−0.286 − 0.957i)16-s + (−7.02 − 2.55i)17-s + ⋯ |
| L(s) = 1 | + (0.631 − 0.317i)2-s + (0.872 − 0.488i)3-s + (0.298 − 0.401i)4-s + (−0.169 + 0.566i)5-s + (0.396 − 0.585i)6-s + (−0.290 + 0.673i)7-s + (0.0613 − 0.348i)8-s + (0.523 − 0.851i)9-s + (0.0726 + 0.412i)10-s + (−0.635 − 0.150i)11-s + (0.0648 − 0.495i)12-s + (−0.471 + 0.309i)13-s + (0.0301 + 0.517i)14-s + (0.128 + 0.577i)15-s + (−0.0717 − 0.239i)16-s + (−1.70 − 0.619i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 + 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.826 + 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.75815 - 0.542027i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.75815 - 0.542027i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.893 + 0.448i)T \) |
| 3 | \( 1 + (-1.51 + 0.845i)T \) |
| good | 5 | \( 1 + (0.379 - 1.26i)T + (-4.17 - 2.74i)T^{2} \) |
| 7 | \( 1 + (0.768 - 1.78i)T + (-4.80 - 5.09i)T^{2} \) |
| 11 | \( 1 + (2.10 + 0.499i)T + (9.82 + 4.93i)T^{2} \) |
| 13 | \( 1 + (1.69 - 1.11i)T + (5.14 - 11.9i)T^{2} \) |
| 17 | \( 1 + (7.02 + 2.55i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (-2.13 + 0.776i)T + (14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (-1.34 - 3.12i)T + (-15.7 + 16.7i)T^{2} \) |
| 29 | \( 1 + (0.282 - 4.85i)T + (-28.8 - 3.36i)T^{2} \) |
| 31 | \( 1 + (-5.07 - 0.593i)T + (30.1 + 7.14i)T^{2} \) |
| 37 | \( 1 + (-4.70 + 3.95i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (6.98 + 3.50i)T + (24.4 + 32.8i)T^{2} \) |
| 43 | \( 1 + (4.77 - 5.06i)T + (-2.50 - 42.9i)T^{2} \) |
| 47 | \( 1 + (-4.18 + 0.489i)T + (45.7 - 10.8i)T^{2} \) |
| 53 | \( 1 + (5.39 + 9.34i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.72 + 1.59i)T + (52.7 - 26.4i)T^{2} \) |
| 61 | \( 1 + (5.37 + 7.22i)T + (-17.4 + 58.4i)T^{2} \) |
| 67 | \( 1 + (-0.661 - 11.3i)T + (-66.5 + 7.77i)T^{2} \) |
| 71 | \( 1 + (-0.518 - 2.93i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-1.80 + 10.2i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-9.16 + 4.60i)T + (47.1 - 63.3i)T^{2} \) |
| 83 | \( 1 + (-3.24 + 1.62i)T + (49.5 - 66.5i)T^{2} \) |
| 89 | \( 1 + (-1.42 + 8.10i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (2.30 + 7.70i)T + (-81.0 + 53.3i)T^{2} \) |
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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
−12.97451261709878170533866314158, −11.92648594114697248770163198876, −11.00164397893555184481316029717, −9.662224409704637151712914945783, −8.748880957263855169509662424950, −7.33634388920758710470151894067, −6.51779014768770218148427810369, −4.92133680838536423851680710945, −3.26663303651104179350910083746, −2.33405441117148183421818350509,
2.63315780385185401385809354712, 4.13645821561245840735979459930, 4.91625471803565595118624520757, 6.65072608839464935183486837356, 7.84755644307340192056579291835, 8.667280371216183278436065813415, 9.948613477604685201377437716990, 10.84794429737321200813015794386, 12.30754034306689090318410998372, 13.31077528081622393229804198323
