L(s) = 1 | + (−0.955 − 0.294i)5-s + (0.0747 + 0.997i)11-s + (0.0747 + 0.997i)13-s + (−0.365 − 0.930i)17-s + (0.5 + 0.866i)19-s + (−0.988 + 0.149i)23-s + (0.826 + 0.563i)25-s + (−0.365 − 0.930i)29-s − 31-s + (−0.988 − 0.149i)37-s + (0.733 − 0.680i)41-s + (0.733 + 0.680i)43-s + (−0.900 − 0.433i)47-s + (0.988 − 0.149i)53-s + (0.222 − 0.974i)55-s + ⋯ |
L(s) = 1 | + (−0.955 − 0.294i)5-s + (0.0747 + 0.997i)11-s + (0.0747 + 0.997i)13-s + (−0.365 − 0.930i)17-s + (0.5 + 0.866i)19-s + (−0.988 + 0.149i)23-s + (0.826 + 0.563i)25-s + (−0.365 − 0.930i)29-s − 31-s + (−0.988 − 0.149i)37-s + (0.733 − 0.680i)41-s + (0.733 + 0.680i)43-s + (−0.900 − 0.433i)47-s + (0.988 − 0.149i)53-s + (0.222 − 0.974i)55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.001498855127 + 0.01058284259i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.001498855127 + 0.01058284259i\) |
\(L(1)\) |
\(\approx\) |
\(0.7238266255 + 0.03658385201i\) |
\(L(1)\) |
\(\approx\) |
\(0.7238266255 + 0.03658385201i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.955 - 0.294i)T \) |
| 11 | \( 1 + (0.0747 + 0.997i)T \) |
| 13 | \( 1 + (0.0747 + 0.997i)T \) |
| 17 | \( 1 + (-0.365 - 0.930i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.988 + 0.149i)T \) |
| 29 | \( 1 + (-0.365 - 0.930i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (-0.988 - 0.149i)T \) |
| 41 | \( 1 + (0.733 - 0.680i)T \) |
| 43 | \( 1 + (0.733 + 0.680i)T \) |
| 47 | \( 1 + (-0.900 - 0.433i)T \) |
| 53 | \( 1 + (0.988 - 0.149i)T \) |
| 59 | \( 1 + (-0.222 + 0.974i)T \) |
| 61 | \( 1 + (0.623 - 0.781i)T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (0.623 + 0.781i)T \) |
| 73 | \( 1 + (0.0747 - 0.997i)T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + (0.0747 - 0.997i)T \) |
| 89 | \( 1 + (-0.826 - 0.563i)T \) |
| 97 | \( 1 + (-0.5 + 0.866i)T \) |
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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
−20.355713361792099149997533485259, −19.76942654648664623721153677757, −19.27304920384705383861982177047, −18.31648777969706080900638685278, −17.81634462547351034897675485676, −16.77279785777344158821454167219, −16.03234269533771775180564640882, −15.48773926103771533524149742746, −14.716811921198885021881384233, −13.97601943721509863555047172278, −13.01308653659723724599551614086, −12.4156027693495185158692391651, −11.42428698129937536237396286025, −10.919968191600791005411477540301, −10.24100521200239611160838229174, −9.017599390178394160512873727079, −8.378641654951001419811355013550, −7.67910447489969135929074421900, −6.85883261758239695494379810393, −5.93116406405755721578577058292, −5.13809965863961210906181828604, −3.98838961683355384177323574423, −3.424276644844034557267990589751, −2.557056326606517916866366785730, −1.17877867201388416071477039221,
0.00410438704413486557908908289, 1.48314741963188064319879851149, 2.35625080000399716157304018881, 3.66040780327667410580407740120, 4.20517535513190344789752848301, 5.009431888995527384510058810157, 6.01459855166666008965797900491, 7.18564134789146162451565447006, 7.466361798992200078364725845022, 8.501985704138387278328048873578, 9.3081656151133347809972840647, 9.96578211648945794028136042486, 11.05981299710946107201091780832, 11.82148460527329977927941030638, 12.196353834741890790123805944309, 13.13214641670986267056944713863, 14.07893055954579621549810929148, 14.69813036476492117034422100141, 15.62694703642471774756090128683, 16.14281118809375426528644574951, 16.8001957781989687202463805580, 17.809839009595698678426164806462, 18.44327080878244664384620773371, 19.25655522907888276684387508528, 19.91605836708053941255732414221