L(s) = 1 | + (−0.984 + 0.173i)2-s + (0.642 + 0.766i)3-s + (0.939 − 0.342i)4-s + (−0.766 − 0.642i)6-s + (−0.866 − 0.5i)7-s + (−0.866 + 0.5i)8-s + (−0.173 + 0.984i)9-s + (0.866 + 0.5i)12-s + (0.642 − 0.766i)13-s + (0.939 + 0.342i)14-s + (0.766 − 0.642i)16-s + (−0.984 + 0.173i)17-s − i·18-s + (−0.173 − 0.984i)21-s + (−0.342 − 0.939i)23-s + (−0.939 − 0.342i)24-s + ⋯ |
L(s) = 1 | + (−0.984 + 0.173i)2-s + (0.642 + 0.766i)3-s + (0.939 − 0.342i)4-s + (−0.766 − 0.642i)6-s + (−0.866 − 0.5i)7-s + (−0.866 + 0.5i)8-s + (−0.173 + 0.984i)9-s + (0.866 + 0.5i)12-s + (0.642 − 0.766i)13-s + (0.939 + 0.342i)14-s + (0.766 − 0.642i)16-s + (−0.984 + 0.173i)17-s − i·18-s + (−0.173 − 0.984i)21-s + (−0.342 − 0.939i)23-s + (−0.939 − 0.342i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0148 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0148 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3026825923 - 0.3072027372i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3026825923 - 0.3072027372i\) |
\(L(1)\) |
\(\approx\) |
\(0.6421976249 + 0.08425023052i\) |
\(L(1)\) |
\(\approx\) |
\(0.6421976249 + 0.08425023052i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.984 + 0.173i)T \) |
| 3 | \( 1 + (0.642 + 0.766i)T \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
| 13 | \( 1 + (0.642 - 0.766i)T \) |
| 17 | \( 1 + (-0.984 + 0.173i)T \) |
| 23 | \( 1 + (-0.342 - 0.939i)T \) |
| 29 | \( 1 + (0.173 - 0.984i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (-0.766 + 0.642i)T \) |
| 43 | \( 1 + (-0.342 + 0.939i)T \) |
| 47 | \( 1 + (-0.984 - 0.173i)T \) |
| 53 | \( 1 + (-0.342 - 0.939i)T \) |
| 59 | \( 1 + (-0.173 - 0.984i)T \) |
| 61 | \( 1 + (0.939 - 0.342i)T \) |
| 67 | \( 1 + (-0.984 - 0.173i)T \) |
| 71 | \( 1 + (-0.939 - 0.342i)T \) |
| 73 | \( 1 + (-0.642 - 0.766i)T \) |
| 79 | \( 1 + (0.766 - 0.642i)T \) |
| 83 | \( 1 + (0.866 + 0.5i)T \) |
| 89 | \( 1 + (-0.766 - 0.642i)T \) |
| 97 | \( 1 + (0.984 - 0.173i)T \) |
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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
−21.58754203025158478750606628031, −20.58810017192378981622843943161, −20.00760679983967315844715325542, −19.27281018892976362389541439947, −18.678591554142348790327892599231, −18.125429753741119520460628914902, −17.25795640614505080775784437870, −16.28798939699968109103108802982, −15.59259989028158415463075968475, −14.8562467240185187143508965543, −13.60090814278387756890500956083, −13.07376447369569354036084345562, −12.03448499303975626831013737829, −11.54351410787359236511960697997, −10.38546705445730418384268125617, −9.33964857555419332310161315070, −8.969433991976889514879154819769, −8.17285950545465702567835735114, −7.09460623511647646539529710098, −6.615751765028287528315390558236, −5.74109402520002946014833670503, −3.89540486650991995473082355313, −3.0670664428473385453752127826, −2.17004746165161684463573856637, −1.34264541940747586818868203039,
0.22124769803328617619746999878, 1.80392673999584444033590749553, 2.87657244200428769228164626778, 3.614433344866999944150685436657, 4.77059956371544664326168394467, 6.01978800506258351046353298712, 6.75122914213044449935969246167, 7.85055037197312509609758474757, 8.498862520078882541722166164799, 9.28704739923670396412707611161, 10.07395656458298568447237620495, 10.611824628147152759465843345144, 11.37862031537337788705222138111, 12.73514728449359224964986036995, 13.50057794274579824890509973055, 14.54464145703096109567796367109, 15.270617070598025110464753127679, 16.1187680507647584477811026123, 16.35833748011061370192160510598, 17.46078713509994527668151011146, 18.18519501574042258692577847113, 19.249955658998977060378686847389, 19.74000879727630318378849021958, 20.3812359520981834287793921508, 21.02127041597851823776364226138