| L(s) = 1 | + (−1.75 − 0.153i)2-s + (0.305 + 1.70i)3-s + (1.09 + 0.192i)4-s + (0.131 − 2.23i)5-s + (−0.274 − 3.04i)6-s + (2.75 + 1.93i)7-s + (1.51 + 0.406i)8-s + (−2.81 + 1.04i)9-s + (−0.574 + 3.90i)10-s + (−1.11 + 3.07i)11-s + (0.00542 + 1.92i)12-s + (0.506 + 5.79i)13-s + (−4.55 − 3.81i)14-s + (3.84 − 0.457i)15-s + (−4.68 − 1.70i)16-s + (1.11 − 0.298i)17-s + ⋯ |
| L(s) = 1 | + (−1.24 − 0.108i)2-s + (0.176 + 0.984i)3-s + (0.546 + 0.0963i)4-s + (0.0589 − 0.998i)5-s + (−0.112 − 1.24i)6-s + (1.04 + 0.730i)7-s + (0.536 + 0.143i)8-s + (−0.937 + 0.347i)9-s + (−0.181 + 1.23i)10-s + (−0.337 + 0.926i)11-s + (0.00156 + 0.554i)12-s + (0.140 + 1.60i)13-s + (−1.21 − 1.02i)14-s + (0.993 − 0.118i)15-s + (−1.17 − 0.426i)16-s + (0.270 − 0.0723i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.519 - 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.519 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.568378 + 0.319740i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.568378 + 0.319740i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.305 - 1.70i)T \) |
| 5 | \( 1 + (-0.131 + 2.23i)T \) |
| good | 2 | \( 1 + (1.75 + 0.153i)T + (1.96 + 0.347i)T^{2} \) |
| 7 | \( 1 + (-2.75 - 1.93i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (1.11 - 3.07i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.506 - 5.79i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-1.11 + 0.298i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-6.40 + 3.69i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.09 - 1.56i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-0.381 + 0.320i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.573 + 3.25i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (1.17 + 4.40i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (1.39 - 1.65i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (1.68 + 3.62i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (2.84 - 4.06i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-1.25 + 1.25i)T - 53iT^{2} \) |
| 59 | \( 1 + (-9.92 + 3.61i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (1.82 + 10.3i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (5.82 - 0.509i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (6.62 + 3.82i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (1.84 - 6.89i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (5.08 + 6.06i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.544 + 6.22i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-0.260 - 0.450i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.27 + 1.99i)T + (62.3 - 74.3i)T^{2} \) |
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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
−13.54678464711541758641290473056, −11.83449311471002827344423053688, −11.29329337524237548389852428902, −9.838104713451420948462166860867, −9.254354285677039563931604524458, −8.570997673819825432563639376310, −7.51394537622064924599052745778, −5.21750033843364816841701510368, −4.51486232603673852332257930223, −1.91717509094390598938957135849,
1.14740426391841459494565543638, 3.16290803719521672587603140349, 5.62183929553488079141814559454, 7.12944071467504049465799647089, 7.85833863782783235911238031603, 8.421047061489202909886927644288, 10.11463118850629163433042859665, 10.77529397531334338600695957768, 11.72532911748182089962350436501, 13.32334161742415727801117325330
