L(s) = 1 | + (0.970 + 0.239i)2-s + (0.885 + 0.464i)4-s + (−0.996 − 0.0804i)5-s + (0.748 + 0.663i)8-s + (−0.948 − 0.316i)10-s + (−0.692 − 0.721i)11-s + (0.568 + 0.822i)16-s + (−0.748 − 0.663i)17-s + (0.5 − 0.866i)19-s + (−0.845 − 0.534i)20-s + (−0.5 − 0.866i)22-s − 23-s + (0.987 + 0.160i)25-s + (−0.692 + 0.721i)29-s + (0.632 + 0.774i)31-s + (0.354 + 0.935i)32-s + ⋯ |
L(s) = 1 | + (0.970 + 0.239i)2-s + (0.885 + 0.464i)4-s + (−0.996 − 0.0804i)5-s + (0.748 + 0.663i)8-s + (−0.948 − 0.316i)10-s + (−0.692 − 0.721i)11-s + (0.568 + 0.822i)16-s + (−0.748 − 0.663i)17-s + (0.5 − 0.866i)19-s + (−0.845 − 0.534i)20-s + (−0.5 − 0.866i)22-s − 23-s + (0.987 + 0.160i)25-s + (−0.692 + 0.721i)29-s + (0.632 + 0.774i)31-s + (0.354 + 0.935i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3845380441 + 1.079463342i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3845380441 + 1.079463342i\) |
\(L(1)\) |
\(\approx\) |
\(1.282913278 + 0.2900756520i\) |
\(L(1)\) |
\(\approx\) |
\(1.282913278 + 0.2900756520i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.970 + 0.239i)T \) |
| 5 | \( 1 + (-0.996 - 0.0804i)T \) |
| 11 | \( 1 + (-0.692 - 0.721i)T \) |
| 17 | \( 1 + (-0.748 - 0.663i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (-0.692 + 0.721i)T \) |
| 31 | \( 1 + (0.632 + 0.774i)T \) |
| 37 | \( 1 + (-0.354 + 0.935i)T \) |
| 41 | \( 1 + (0.799 + 0.600i)T \) |
| 43 | \( 1 + (-0.632 + 0.774i)T \) |
| 47 | \( 1 + (-0.845 - 0.534i)T \) |
| 53 | \( 1 + (-0.948 + 0.316i)T \) |
| 59 | \( 1 + (0.568 - 0.822i)T \) |
| 61 | \( 1 + (0.200 + 0.979i)T \) |
| 67 | \( 1 + (-0.845 - 0.534i)T \) |
| 71 | \( 1 + (0.919 - 0.391i)T \) |
| 73 | \( 1 + (-0.278 + 0.960i)T \) |
| 79 | \( 1 + (-0.845 - 0.534i)T \) |
| 83 | \( 1 + (0.120 + 0.992i)T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + (-0.428 + 0.903i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.68134375991559920581896979520, −17.82850580584430907919075429049, −16.918050678128906616896911397089, −15.93063032980908703667705250781, −15.74592824290220768353309609215, −14.93769090312545492850799318390, −14.43045307945932240069538196557, −13.52235480573377645217986271608, −12.83553720682106536789157664979, −12.25194027571446839832164580880, −11.6478729116915482930885232362, −10.94218001419918728759795907147, −10.272308236644947633626334694868, −9.56012142160482323808941671975, −8.29096378917283719666381491582, −7.7084774034482046050748086543, −7.0877911629896427891383249920, −6.16257233244762894877629303120, −5.48927546606800665048738452519, −4.50461153443902439675239738951, −4.06015652623608047961593126940, −3.3297006510432957248826090290, −2.3439373558192558969431676141, −1.68615975702172688383320419955, −0.22527572932199385260888476878,
1.17883680513824296365004038894, 2.46404792877002679098571668402, 3.12442451884736840061033965855, 3.76044487301053035206332881757, 4.83505676890198456506524499527, 5.01071619662005678189791873093, 6.18440393506878628314416105549, 6.8378566269168123879726429527, 7.60506011093658820914753802915, 8.19163098050358054906829666090, 8.91103500451962837793691695081, 10.08065517961904841335637470995, 11.03780358787355459631496754729, 11.396150539882240750363803929889, 12.051726767482685887927484056294, 12.88620272785422224377651221734, 13.44540785854257917064841571193, 14.11981445654890340739677459542, 14.92029178168882651310467852282, 15.60742250322933218712612032610, 16.09141904502490667060953192317, 16.48213281172637808105841696420, 17.59587424015443574620954013023, 18.27555385614645897025380452427, 19.15338248508012693613458596204