L(s) = 1 | + (−0.579 − 0.815i)2-s + (−0.737 + 0.675i)3-s + (−0.329 + 0.944i)4-s + (0.911 − 0.411i)5-s + (0.977 + 0.210i)6-s + (−0.925 + 0.378i)7-s + (0.960 − 0.278i)8-s + (0.0881 − 0.996i)9-s + (−0.863 − 0.505i)10-s + (0.158 − 0.987i)11-s + (−0.394 − 0.918i)12-s + (−0.261 + 0.965i)13-s + (0.844 + 0.535i)14-s + (−0.394 + 0.918i)15-s + (−0.783 − 0.621i)16-s + (0.427 + 0.904i)17-s + ⋯ |
L(s) = 1 | + (−0.579 − 0.815i)2-s + (−0.737 + 0.675i)3-s + (−0.329 + 0.944i)4-s + (0.911 − 0.411i)5-s + (0.977 + 0.210i)6-s + (−0.925 + 0.378i)7-s + (0.960 − 0.278i)8-s + (0.0881 − 0.996i)9-s + (−0.863 − 0.505i)10-s + (0.158 − 0.987i)11-s + (−0.394 − 0.918i)12-s + (−0.261 + 0.965i)13-s + (0.844 + 0.535i)14-s + (−0.394 + 0.918i)15-s + (−0.783 − 0.621i)16-s + (0.427 + 0.904i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6854696628 + 0.02151119179i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6854696628 + 0.02151119179i\) |
\(L(1)\) |
\(\approx\) |
\(0.6892445959 - 0.06019923386i\) |
\(L(1)\) |
\(\approx\) |
\(0.6892445959 - 0.06019923386i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 179 | \( 1 \) |
good | 2 | \( 1 + (-0.579 - 0.815i)T \) |
| 3 | \( 1 + (-0.737 + 0.675i)T \) |
| 5 | \( 1 + (0.911 - 0.411i)T \) |
| 7 | \( 1 + (-0.925 + 0.378i)T \) |
| 11 | \( 1 + (0.158 - 0.987i)T \) |
| 13 | \( 1 + (-0.261 + 0.965i)T \) |
| 17 | \( 1 + (0.427 + 0.904i)T \) |
| 19 | \( 1 + (0.362 + 0.932i)T \) |
| 23 | \( 1 + (0.990 - 0.140i)T \) |
| 29 | \( 1 + (0.804 - 0.593i)T \) |
| 31 | \( 1 + (-0.825 + 0.564i)T \) |
| 37 | \( 1 + (0.804 + 0.593i)T \) |
| 41 | \( 1 + (0.997 + 0.0705i)T \) |
| 43 | \( 1 + (0.662 + 0.749i)T \) |
| 47 | \( 1 + (-0.969 + 0.244i)T \) |
| 53 | \( 1 + (0.607 - 0.794i)T \) |
| 59 | \( 1 + (-0.123 + 0.992i)T \) |
| 61 | \( 1 + (0.880 - 0.474i)T \) |
| 67 | \( 1 + (0.960 + 0.278i)T \) |
| 71 | \( 1 + (-0.949 + 0.312i)T \) |
| 73 | \( 1 + (-0.999 - 0.0352i)T \) |
| 79 | \( 1 + (0.489 + 0.871i)T \) |
| 83 | \( 1 + (-0.635 - 0.772i)T \) |
| 89 | \( 1 + (-0.579 + 0.815i)T \) |
| 97 | \( 1 + (-0.520 - 0.854i)T \) |
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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
−27.34738106830853142540872106131, −26.13328055911850712315079383909, −25.2696239293651687296677297084, −24.84914371346412035268111452037, −23.43598018338088974063962287282, −22.76990077895911140662267261639, −22.10846276615915178029445233477, −20.24242909332096813627907673748, −19.2847620478091528486263746767, −18.19079249942986279168405032798, −17.64365540485423422240151251461, −16.83073874614968468078931711945, −15.81356789440787557556625313240, −14.561589515078221989896942350560, −13.44815095262612305248543326117, −12.70686902996700985344805762176, −11.00229801869098994360183953761, −10.06332238494196698625641383314, −9.26713603871675177430935995399, −7.37564932370555131720179284619, −6.988048201491860871921476640224, −5.85775492186319002713751967560, −4.963044476276446831569002804, −2.56074188384225755954630290059, −0.90680519509414205430829416921,
1.21265527684763347341323669663, 2.92616584915091246267999093203, 4.14243934957180682439844895270, 5.5887678488197216262834992846, 6.57635205659634454718753910336, 8.57190121149098556920366081304, 9.46193902453005547329265352392, 10.10128466526891268872494943444, 11.21238712874573840256384423529, 12.27237766470375844625130643098, 13.0837863259149134107354541102, 14.42091059490419597431598429431, 16.33370640741234409950601482775, 16.51615029524837626139553376300, 17.56389554123066681677320204476, 18.66379956552213643968172379787, 19.51525282705304038147066580339, 20.94895563764452149584892757197, 21.48442484953369152206851379019, 22.12891700081553954225714938974, 23.20165815545137930394918894507, 24.66583641672081478450669957364, 25.79252957042090570764682642928, 26.57113265923526653870967560585, 27.51185735120832720610598065014