L(s) = 1 | + (−1.28 − 0.599i)2-s + (1.28 + 1.53i)4-s + i·5-s + (−2.60 − 0.468i)7-s + (−0.719 − 2.73i)8-s + (0.599 − 1.28i)10-s + 2.39i·11-s − 2i·13-s + (3.05 + 2.16i)14-s + (−0.719 + 3.93i)16-s − 7.12i·17-s + 2.39·19-s + (−1.53 + 1.28i)20-s + (1.43 − 3.07i)22-s + 5.73i·23-s + ⋯ |
L(s) = 1 | + (−0.905 − 0.424i)2-s + (0.640 + 0.768i)4-s + 0.447i·5-s + (−0.984 − 0.176i)7-s + (−0.254 − 0.967i)8-s + (0.189 − 0.405i)10-s + 0.723i·11-s − 0.554i·13-s + (0.816 + 0.577i)14-s + (−0.179 + 0.983i)16-s − 1.72i·17-s + 0.550·19-s + (−0.343 + 0.286i)20-s + (0.306 − 0.654i)22-s + 1.19i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8427333384\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8427333384\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.28 + 0.599i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (2.60 + 0.468i)T \) |
good | 11 | \( 1 - 2.39iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 7.12iT - 17T^{2} \) |
| 19 | \( 1 - 2.39T + 19T^{2} \) |
| 23 | \( 1 - 5.73iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 6.67T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 7.12iT - 41T^{2} \) |
| 43 | \( 1 - 7.60iT - 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 - 14.2iT - 67T^{2} \) |
| 71 | \( 1 - 6.14iT - 71T^{2} \) |
| 73 | \( 1 - 9.36iT - 73T^{2} \) |
| 79 | \( 1 - 4.27iT - 79T^{2} \) |
| 83 | \( 1 + 0.936T + 83T^{2} \) |
| 89 | \( 1 - 12iT - 89T^{2} \) |
| 97 | \( 1 + 7.12iT - 97T^{2} \) |
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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
−9.681562319142330563847349408795, −9.350327605476777763919457697425, −8.088069863639570539939610923839, −7.32087594161156062045365215294, −6.81943783854345252691036265575, −5.77284202354776417444924977673, −4.41547925540822320164062646845, −3.14405562969957160935382376387, −2.66752042686840965007719339485, −0.981807626920474700063787404827,
0.61115613827000443558099971085, 2.05583407559359198429732609538, 3.32201745632153607625969098917, 4.57439769530577810124696185145, 5.91955005660516793155413640944, 6.21211928860511017218408164843, 7.19710930969244880751337395190, 8.234313847696796411172175872079, 8.781972969937951339482284147711, 9.424736346998069908328149488456