| L(s) = 1 | + (0.178 + 1.40i)2-s + (−1.93 + 0.5i)4-s − 2.80i·5-s − 7-s + (−1.04 − 2.62i)8-s + (3.93 − 0.5i)10-s + 3.51i·11-s − 4.87i·13-s + (−0.178 − 1.40i)14-s + (3.50 − 1.93i)16-s − 3.16·17-s + 7.87i·19-s + (1.40 + 5.43i)20-s + (−4.93 + 0.627i)22-s + 0.356·23-s + ⋯ |
| L(s) = 1 | + (0.126 + 0.992i)2-s + (−0.968 + 0.250i)4-s − 1.25i·5-s − 0.377·7-s + (−0.370 − 0.929i)8-s + (1.24 − 0.158i)10-s + 1.06i·11-s − 1.35i·13-s + (−0.0476 − 0.374i)14-s + (0.875 − 0.484i)16-s − 0.766·17-s + 1.80i·19-s + (0.313 + 1.21i)20-s + (−1.05 + 0.133i)22-s + 0.0743·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.370i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2277012247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2277012247\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.178 - 1.40i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 2.80iT - 5T^{2} \) |
| 11 | \( 1 - 3.51iT - 11T^{2} \) |
| 13 | \( 1 + 4.87iT - 13T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 19 | \( 1 - 7.87iT - 19T^{2} \) |
| 23 | \( 1 - 0.356T + 23T^{2} \) |
| 29 | \( 1 + 2.44iT - 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 - 5.87iT - 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 8.87iT - 43T^{2} \) |
| 47 | \( 1 + 7.34T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 12.9iT - 59T^{2} \) |
| 61 | \( 1 - 1.74iT - 61T^{2} \) |
| 67 | \( 1 - 14.6iT - 67T^{2} \) |
| 71 | \( 1 + 3.51T + 71T^{2} \) |
| 73 | \( 1 + 6.87T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 + 12.9iT - 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 - 5.74T + 97T^{2} \) |
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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
−9.863118676080488195895813808743, −8.899216252980531808950738894637, −8.225725630733890237174614893146, −7.65667012239835563533173047903, −6.64529158044526215955296126782, −5.75387321593438257254833489371, −5.06238432569001007107770449209, −4.32146073848751213857063160844, −3.31186567848027221997734856023, −1.50082046866480656820952361138,
0.085707919404388907742533411820, 1.93738935616731485086851826324, 2.88300001516468106532973696540, 3.58498268007406818316704038461, 4.57688917594503495692754608384, 5.66904162465330693402054796891, 6.69303260497936136067645421126, 7.16652674968239792617921100744, 8.731873623298419456057390361685, 9.003243657274397728628765994553
