L(s) = 1 | + (−0.766 + 0.642i)2-s + (−0.993 − 0.116i)3-s + (0.173 − 0.984i)4-s + (0.286 − 0.957i)5-s + (0.835 − 0.549i)6-s + (−0.686 − 0.727i)7-s + (0.5 + 0.866i)8-s + (0.973 + 0.230i)9-s + (0.396 + 0.918i)10-s + (0.396 − 0.918i)11-s + (−0.286 + 0.957i)12-s + (0.686 + 0.727i)13-s + (0.993 + 0.116i)14-s + (−0.396 + 0.918i)15-s + (−0.939 − 0.342i)16-s + (0.939 + 0.342i)17-s + ⋯ |
L(s) = 1 | + (−0.766 + 0.642i)2-s + (−0.993 − 0.116i)3-s + (0.173 − 0.984i)4-s + (0.286 − 0.957i)5-s + (0.835 − 0.549i)6-s + (−0.686 − 0.727i)7-s + (0.5 + 0.866i)8-s + (0.973 + 0.230i)9-s + (0.396 + 0.918i)10-s + (0.396 − 0.918i)11-s + (−0.286 + 0.957i)12-s + (0.686 + 0.727i)13-s + (0.993 + 0.116i)14-s + (−0.396 + 0.918i)15-s + (−0.939 − 0.342i)16-s + (0.939 + 0.342i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7505020484 - 0.3244901272i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7505020484 - 0.3244901272i\) |
\(L(1)\) |
\(\approx\) |
\(0.6011407493 - 0.05181777700i\) |
\(L(1)\) |
\(\approx\) |
\(0.6011407493 - 0.05181777700i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 3 | \( 1 + (-0.993 - 0.116i)T \) |
| 5 | \( 1 + (0.286 - 0.957i)T \) |
| 7 | \( 1 + (-0.686 - 0.727i)T \) |
| 11 | \( 1 + (0.396 - 0.918i)T \) |
| 13 | \( 1 + (0.686 + 0.727i)T \) |
| 17 | \( 1 + (0.939 + 0.342i)T \) |
| 19 | \( 1 + (-0.766 + 0.642i)T \) |
| 23 | \( 1 + (0.939 - 0.342i)T \) |
| 29 | \( 1 + (-0.597 + 0.802i)T \) |
| 31 | \( 1 + (0.286 + 0.957i)T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (-0.173 - 0.984i)T \) |
| 47 | \( 1 + (0.893 - 0.448i)T \) |
| 53 | \( 1 + (-0.686 - 0.727i)T \) |
| 59 | \( 1 + (-0.396 + 0.918i)T \) |
| 61 | \( 1 + (0.835 - 0.549i)T \) |
| 67 | \( 1 + (0.973 + 0.230i)T \) |
| 71 | \( 1 + (0.766 + 0.642i)T \) |
| 73 | \( 1 + (0.396 - 0.918i)T \) |
| 79 | \( 1 + (0.286 + 0.957i)T \) |
| 83 | \( 1 + (0.396 + 0.918i)T \) |
| 89 | \( 1 + (-0.893 - 0.448i)T \) |
| 97 | \( 1 + (0.686 + 0.727i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.69967309842765080559373397834, −17.943878968754577642966504494239, −17.24235594548163077767480187948, −16.92800182864075117214041493791, −15.81815236759353232859780338237, −15.4152102672425317804744873614, −14.73462927981452947980128677504, −13.32214539401300607086328728899, −12.96171446917098735492089556679, −12.19001232523947898141451567179, −11.51282123222517472820701511231, −10.9974586686501903756555683734, −10.29347375432328055209656231362, −9.5729004096008426975582805938, −9.361701927194175343649159665984, −8.04627657086909480601891882171, −7.34338167803388884155724830550, −6.55660017047469167300514782092, −6.127924372282958750937025239324, −5.16365578967348536328421407917, −4.11991690869182639519408783992, −3.31487672858251184262994474069, −2.604821759136470547172529212144, −1.74486545613112691174850129423, −0.72047232688028167117898061294,
0.56824438236847061861235084504, 1.17900466659602642673096285315, 1.85915287876052901065642654710, 3.564321803909688037311216846500, 4.25788451991864174619294160595, 5.32576612127232896582747171226, 5.6769183892554276572674572280, 6.6438863881075724706115744600, 6.82506088672289304176696120693, 7.97856930928945130108276526528, 8.67838419299911605772396479608, 9.282829488606075180401810504084, 10.13177409320100647890727243240, 10.64853129591329824572503868301, 11.297456011335825207684811996438, 12.25828070639068300959367357106, 12.831582422492577488086913983928, 13.738791078831711007075677665798, 14.152102893780041317499007596474, 15.32146027324084485300641097297, 16.147139752804830013858246633719, 16.5132492472041815017486059121, 16.953742568966014732397717915757, 17.29619158753643292409531001727, 18.386217613013054357230499053819