| L(s) = 1 | + (0.908 − 1.08i)2-s + (−0.347 − 1.96i)4-s − 0.863·5-s + i·7-s + (−2.44 − 1.41i)8-s + (−0.784 + 0.935i)10-s − 2.62i·11-s + 1.01i·13-s + (1.08 + 0.908i)14-s + (−3.75 + 1.36i)16-s − 2.34i·17-s − 4.09·19-s + (0.300 + 1.70i)20-s + (−2.84 − 2.39i)22-s − 6.23·23-s + ⋯ |
| L(s) = 1 | + (0.642 − 0.766i)2-s + (−0.173 − 0.984i)4-s − 0.386·5-s + 0.377i·7-s + (−0.866 − 0.499i)8-s + (−0.248 + 0.295i)10-s − 0.792i·11-s + 0.282i·13-s + (0.289 + 0.242i)14-s + (−0.939 + 0.342i)16-s − 0.568i·17-s − 0.939·19-s + (0.0671 + 0.380i)20-s + (−0.607 − 0.509i)22-s − 1.29·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7960023293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7960023293\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.908 + 1.08i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 5 | \( 1 + 0.863T + 5T^{2} \) |
| 11 | \( 1 + 2.62iT - 11T^{2} \) |
| 13 | \( 1 - 1.01iT - 13T^{2} \) |
| 17 | \( 1 + 2.34iT - 17T^{2} \) |
| 19 | \( 1 + 4.09T + 19T^{2} \) |
| 23 | \( 1 + 6.23T + 23T^{2} \) |
| 29 | \( 1 - 1.57T + 29T^{2} \) |
| 31 | \( 1 + 6.29iT - 31T^{2} \) |
| 37 | \( 1 - 5.18iT - 37T^{2} \) |
| 41 | \( 1 + 10.1iT - 41T^{2} \) |
| 43 | \( 1 + 0.496T + 43T^{2} \) |
| 47 | \( 1 - 5.24T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 - 4.40iT - 59T^{2} \) |
| 61 | \( 1 - 9.59iT - 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 - 2.11T + 71T^{2} \) |
| 73 | \( 1 + 8.27T + 73T^{2} \) |
| 79 | \( 1 - 9.44iT - 79T^{2} \) |
| 83 | \( 1 + 6.62iT - 83T^{2} \) |
| 89 | \( 1 + 10.2iT - 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−9.087459906364555695328613707474, −8.402278310168077421044212501607, −7.36851231124988562220613904179, −6.16850013930314309506548406360, −5.75706658044800625366496555321, −4.54312604676009053778686963214, −3.89515203975427120158349295306, −2.84778984078479610460012223604, −1.86599111130276293517716347142, −0.23462460594835998998298582264,
2.01596892285150184594458666833, 3.36173760636547085889712318288, 4.19879746770197011533347804499, 4.83796734450479557951824107548, 6.00854081269511662186583332677, 6.58065001389222612256672058492, 7.62103904708327774875580350712, 7.999780728442818291463796276654, 8.915897427031890603874012796918, 9.899336974148004814666103935539
