L(s) = 1 | + (−0.837 + 0.546i)2-s + (−0.879 + 0.475i)3-s + (0.401 − 0.915i)4-s + (0.0825 + 0.996i)5-s + (0.475 − 0.879i)6-s + (−0.324 − 0.945i)7-s + (0.164 + 0.986i)8-s + (0.546 − 0.837i)9-s + (−0.614 − 0.789i)10-s + (−0.245 + 0.969i)11-s + (0.0825 + 0.996i)12-s + (0.996 − 0.0825i)13-s + (0.789 + 0.614i)14-s + (−0.546 − 0.837i)15-s + (−0.677 − 0.735i)16-s + (−0.0825 − 0.996i)17-s + ⋯ |
L(s) = 1 | + (−0.837 + 0.546i)2-s + (−0.879 + 0.475i)3-s + (0.401 − 0.915i)4-s + (0.0825 + 0.996i)5-s + (0.475 − 0.879i)6-s + (−0.324 − 0.945i)7-s + (0.164 + 0.986i)8-s + (0.546 − 0.837i)9-s + (−0.614 − 0.789i)10-s + (−0.245 + 0.969i)11-s + (0.0825 + 0.996i)12-s + (0.996 − 0.0825i)13-s + (0.789 + 0.614i)14-s + (−0.546 − 0.837i)15-s + (−0.677 − 0.735i)16-s + (−0.0825 − 0.996i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.08450198258 + 0.1609294894i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.08450198258 + 0.1609294894i\) |
\(L(1)\) |
\(\approx\) |
\(0.4184264306 + 0.2381987279i\) |
\(L(1)\) |
\(\approx\) |
\(0.4184264306 + 0.2381987279i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 229 | \( 1 \) |
good | 2 | \( 1 + (-0.837 + 0.546i)T \) |
| 3 | \( 1 + (-0.879 + 0.475i)T \) |
| 5 | \( 1 + (0.0825 + 0.996i)T \) |
| 7 | \( 1 + (-0.324 - 0.945i)T \) |
| 11 | \( 1 + (-0.245 + 0.969i)T \) |
| 13 | \( 1 + (0.996 - 0.0825i)T \) |
| 17 | \( 1 + (-0.0825 - 0.996i)T \) |
| 19 | \( 1 + (-0.0825 + 0.996i)T \) |
| 23 | \( 1 + (-0.614 + 0.789i)T \) |
| 29 | \( 1 + (0.324 + 0.945i)T \) |
| 31 | \( 1 + (0.969 + 0.245i)T \) |
| 37 | \( 1 + (-0.677 + 0.735i)T \) |
| 41 | \( 1 + (0.837 - 0.546i)T \) |
| 43 | \( 1 + (-0.677 + 0.735i)T \) |
| 47 | \( 1 + (-0.837 - 0.546i)T \) |
| 53 | \( 1 + (-0.879 + 0.475i)T \) |
| 59 | \( 1 + (0.735 - 0.677i)T \) |
| 61 | \( 1 + (0.546 - 0.837i)T \) |
| 67 | \( 1 + (-0.837 - 0.546i)T \) |
| 71 | \( 1 + (-0.245 - 0.969i)T \) |
| 73 | \( 1 + (-0.164 - 0.986i)T \) |
| 79 | \( 1 + (0.324 - 0.945i)T \) |
| 83 | \( 1 + (-0.677 - 0.735i)T \) |
| 89 | \( 1 + iT \) |
| 97 | \( 1 + (0.986 + 0.164i)T \) |
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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
−25.530730297618064338629122264592, −24.56633378280772148641902630529, −23.97427088476571847297621014556, −22.57327230771311324288742574153, −21.547104705568723522560192421097, −21.03619491294106324175099771232, −19.56199799476350324326249603267, −18.97109599771383723212792229574, −17.99009833766664115155877772662, −17.22715922245593799637162158350, −16.17478259361859362491684029594, −15.76299216780930298114701750436, −13.41033385675147442770849911207, −12.81639749147481959979963430563, −11.870921788932726906147078135960, −11.12307878320158288368966714073, −9.97227456489890102129429200796, −8.64838852892744168513776646796, −8.220200358073363742373200264823, −6.49028541087038889793178041396, −5.711655491757648995751027469755, −4.18549103422870350121423988960, −2.47127382997210263974112161450, −1.20076450981105046810199198258, −0.10123989413445595901505765100,
1.469024674497635865482106427196, 3.42779576628467340374688170518, 4.86889205665746998324106045363, 6.19288763369586135541559898681, 6.85951456798284978969201502265, 7.81238463458265199660795799047, 9.551217341875017481822188140420, 10.21918504972845696426776244135, 10.86992002941220096646356422125, 11.914741784458946724754954316555, 13.61751176834508245322543392457, 14.64902449638905825801967978885, 15.73362406368189001179064558017, 16.28042402621934310225039382573, 17.52027688776583371471220962753, 17.97793596804433727270556141690, 18.9065150520608417370065414523, 20.184547699023062981629368344163, 21.01240739969079859781814408264, 22.56052576667928128943400575937, 23.08145891187594477826218682176, 23.691146185360710614615782240392, 25.237253707830247349523619099881, 26.002929910493579037504081486165, 26.74743442546835785098784691107