L(s) = 1 | + (−0.809 + 0.587i)2-s + (−0.913 + 0.406i)3-s + (0.309 − 0.951i)4-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)6-s + (0.669 + 0.743i)7-s + (0.309 + 0.951i)8-s + (0.669 − 0.743i)9-s + (0.913 + 0.406i)10-s + (0.978 − 0.207i)11-s + (0.104 + 0.994i)12-s + (0.104 − 0.994i)13-s + (−0.978 − 0.207i)14-s + (0.809 + 0.587i)15-s + (−0.809 − 0.587i)16-s + (0.978 + 0.207i)17-s + ⋯ |
L(s) = 1 | + (−0.809 + 0.587i)2-s + (−0.913 + 0.406i)3-s + (0.309 − 0.951i)4-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)6-s + (0.669 + 0.743i)7-s + (0.309 + 0.951i)8-s + (0.669 − 0.743i)9-s + (0.913 + 0.406i)10-s + (0.978 − 0.207i)11-s + (0.104 + 0.994i)12-s + (0.104 − 0.994i)13-s + (−0.978 − 0.207i)14-s + (0.809 + 0.587i)15-s + (−0.809 − 0.587i)16-s + (0.978 + 0.207i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7022495243 - 0.05637409470i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7022495243 - 0.05637409470i\) |
\(L(1)\) |
\(\approx\) |
\(0.6202467172 + 0.05985226335i\) |
\(L(1)\) |
\(\approx\) |
\(0.6202467172 + 0.05985226335i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 31 | \( 1 \) |
good | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 3 | \( 1 + (-0.913 + 0.406i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.669 + 0.743i)T \) |
| 11 | \( 1 + (0.978 - 0.207i)T \) |
| 13 | \( 1 + (0.104 - 0.994i)T \) |
| 17 | \( 1 + (0.978 + 0.207i)T \) |
| 19 | \( 1 + (-0.104 - 0.994i)T \) |
| 23 | \( 1 + (-0.309 - 0.951i)T \) |
| 29 | \( 1 + (0.809 - 0.587i)T \) |
| 37 | \( 1 + (0.5 - 0.866i)T \) |
| 41 | \( 1 + (0.913 + 0.406i)T \) |
| 43 | \( 1 + (0.104 + 0.994i)T \) |
| 47 | \( 1 + (-0.809 - 0.587i)T \) |
| 53 | \( 1 + (-0.669 + 0.743i)T \) |
| 59 | \( 1 + (0.913 - 0.406i)T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.669 - 0.743i)T \) |
| 73 | \( 1 + (0.978 - 0.207i)T \) |
| 79 | \( 1 + (0.978 + 0.207i)T \) |
| 83 | \( 1 + (-0.913 - 0.406i)T \) |
| 89 | \( 1 + (-0.309 + 0.951i)T \) |
| 97 | \( 1 + (0.309 - 0.951i)T \) |
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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
−36.319238269800547231247325605061, −35.47503640768184046308857716618, −34.25856992256049194006659899691, −33.51268828389573773948219880226, −30.961941483346572583639281100692, −30.095417344213222757939348948365, −29.26481552316304555329390952142, −27.63606649473705546323285706009, −27.117107115587394421124069905322, −25.51918721661136353710050655569, −23.81008030009574115476912282889, −22.59612313050221347538468062052, −21.28251709821787186217446922844, −19.56737128813688947762187121293, −18.55459192220367026910783856876, −17.39808738251428242890294361764, −16.32940976421178088738407126061, −14.14829751939124552274094388930, −12.0565183581053145854500972254, −11.293537220783265144132886521030, −10.05733132093505013976163558488, −7.79530942240813706616693139156, −6.72642466416705714583734708087, −3.98399601539889279516376515714, −1.42724419321017576941360630715,
0.86734042277587321242882642972, 4.785421228510313257053987873572, 6.058605118724657326368452102786, 8.06276597845191380458038721800, 9.37512357983081562739443310323, 11.064355719876936311520253835049, 12.25255051944786846348513052556, 14.84761695620097159643897551566, 15.960729740727348081646457283196, 17.04817387003324079448239619211, 18.08877903781105312218405693279, 19.69012179489032358081079039430, 21.17091458259031212326590134544, 22.87607989284162025649510252814, 24.13970476949148202062787359738, 25.00399233464050259355059118338, 26.93346691293924097908014392486, 27.90808326625986959516955632933, 28.3064304639656497189162314094, 30.02946690346975909752496985490, 32.16647466958680711173927161117, 32.89363300430623264508359146105, 34.593170698638995805689464858010, 34.826111336646171946774448193861, 36.2522960507084145325954297370