L(s) = 1 | + (0.669 − 0.743i)2-s + (−0.104 − 0.994i)4-s + (−0.5 − 0.866i)7-s + (−0.809 − 0.587i)8-s + (0.669 − 0.743i)11-s + (0.669 + 0.743i)13-s + (−0.978 − 0.207i)14-s + (−0.978 + 0.207i)16-s + (−0.809 − 0.587i)17-s + (−0.809 − 0.587i)19-s + (−0.104 − 0.994i)22-s + (−0.978 − 0.207i)23-s + 26-s + (−0.809 + 0.587i)28-s + (0.913 + 0.406i)29-s + ⋯ |
L(s) = 1 | + (0.669 − 0.743i)2-s + (−0.104 − 0.994i)4-s + (−0.5 − 0.866i)7-s + (−0.809 − 0.587i)8-s + (0.669 − 0.743i)11-s + (0.669 + 0.743i)13-s + (−0.978 − 0.207i)14-s + (−0.978 + 0.207i)16-s + (−0.809 − 0.587i)17-s + (−0.809 − 0.587i)19-s + (−0.104 − 0.994i)22-s + (−0.978 − 0.207i)23-s + 26-s + (−0.809 + 0.587i)28-s + (0.913 + 0.406i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6006756214 - 1.324182057i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6006756214 - 1.324182057i\) |
\(L(1)\) |
\(\approx\) |
\(1.028879006 - 0.8369493514i\) |
\(L(1)\) |
\(\approx\) |
\(1.028879006 - 0.8369493514i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (0.669 - 0.743i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (0.669 - 0.743i)T \) |
| 13 | \( 1 + (0.669 + 0.743i)T \) |
| 17 | \( 1 + (-0.809 - 0.587i)T \) |
| 19 | \( 1 + (-0.809 - 0.587i)T \) |
| 23 | \( 1 + (-0.978 - 0.207i)T \) |
| 29 | \( 1 + (0.913 + 0.406i)T \) |
| 31 | \( 1 + (0.913 - 0.406i)T \) |
| 37 | \( 1 + (0.309 - 0.951i)T \) |
| 41 | \( 1 + (0.669 + 0.743i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + (0.913 + 0.406i)T \) |
| 53 | \( 1 + (-0.809 + 0.587i)T \) |
| 59 | \( 1 + (0.669 + 0.743i)T \) |
| 61 | \( 1 + (0.669 - 0.743i)T \) |
| 67 | \( 1 + (0.913 - 0.406i)T \) |
| 71 | \( 1 + (-0.809 + 0.587i)T \) |
| 73 | \( 1 + (0.309 + 0.951i)T \) |
| 79 | \( 1 + (0.913 + 0.406i)T \) |
| 83 | \( 1 + (-0.104 + 0.994i)T \) |
| 89 | \( 1 + (0.309 + 0.951i)T \) |
| 97 | \( 1 + (0.913 + 0.406i)T \) |
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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
−26.51066313778163550861495940955, −25.37545187608466524282038202678, −25.19937383495800055282085499038, −23.99122928783037866173461427972, −22.9818513741607851905972660150, −22.31437258003690248868874361228, −21.47815431415996956363305857087, −20.409382843249657766232938237521, −19.277633784718601370679520348389, −17.98014754336955062234887735265, −17.311258567745428831921498053297, −16.02774895717290494683060602660, −15.3876660443267222599253529522, −14.54036962770531742956945165954, −13.33071966393091436299381518232, −12.50739906357817815029609422152, −11.70169637999337385813908129125, −10.134115741469915706802945799977, −8.82225199645707363668397127748, −8.04759680653657946118019804497, −6.527136612293987927889796361739, −6.0233799976182667473408296774, −4.63500721336503486194404051262, −3.56142429829231002347948213480, −2.217132283833823592086517560637,
0.88647056984296021228264334178, 2.4487599474775518744667328909, 3.79259785656296315541265177943, 4.48148340366476285503750741240, 6.13260715094846917211461754516, 6.79789244978775369785301929268, 8.65170274515956896147398090475, 9.65001786269676346804547970110, 10.79672302862459015216760992229, 11.476053301529957589863843836298, 12.66789653830284849882623543048, 13.74011345956584242292873041508, 14.11185372644174663080306344666, 15.58904663599516515312559694414, 16.45373050937762896675826073785, 17.73178666213430673033859971730, 18.94627071728709042696826164456, 19.66262574240741326268643112772, 20.460120666628645539219338548263, 21.538272289706539029902323523980, 22.25990608921744217365091544860, 23.285126021742330262402563301299, 23.92379022012345822801732007317, 24.95718910628205374902619815735, 26.27593868478430169536538660800