L(s) = 1 | + (−0.955 + 0.294i)2-s + (0.826 − 0.563i)4-s + (0.623 − 0.781i)5-s + (−0.623 + 0.781i)8-s + (−0.365 + 0.930i)10-s + (0.222 − 0.974i)11-s + (−0.955 + 0.294i)13-s + (0.365 − 0.930i)16-s + (0.826 + 0.563i)17-s + (0.5 − 0.866i)19-s + (0.0747 − 0.997i)20-s + (0.0747 + 0.997i)22-s + (0.900 + 0.433i)23-s + (−0.222 − 0.974i)25-s + (0.826 − 0.563i)26-s + ⋯ |
L(s) = 1 | + (−0.955 + 0.294i)2-s + (0.826 − 0.563i)4-s + (0.623 − 0.781i)5-s + (−0.623 + 0.781i)8-s + (−0.365 + 0.930i)10-s + (0.222 − 0.974i)11-s + (−0.955 + 0.294i)13-s + (0.365 − 0.930i)16-s + (0.826 + 0.563i)17-s + (0.5 − 0.866i)19-s + (0.0747 − 0.997i)20-s + (0.0747 + 0.997i)22-s + (0.900 + 0.433i)23-s + (−0.222 − 0.974i)25-s + (0.826 − 0.563i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8194712144 - 0.4312378379i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8194712144 - 0.4312378379i\) |
\(L(1)\) |
\(\approx\) |
\(0.7897269310 - 0.1286051987i\) |
\(L(1)\) |
\(\approx\) |
\(0.7897269310 - 0.1286051987i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.955 + 0.294i)T \) |
| 5 | \( 1 + (0.623 - 0.781i)T \) |
| 11 | \( 1 + (0.222 - 0.974i)T \) |
| 13 | \( 1 + (-0.955 + 0.294i)T \) |
| 17 | \( 1 + (0.826 + 0.563i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.900 + 0.433i)T \) |
| 29 | \( 1 + (-0.0747 + 0.997i)T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.0747 - 0.997i)T \) |
| 41 | \( 1 + (-0.988 - 0.149i)T \) |
| 43 | \( 1 + (-0.988 + 0.149i)T \) |
| 47 | \( 1 + (0.955 - 0.294i)T \) |
| 53 | \( 1 + (-0.0747 - 0.997i)T \) |
| 59 | \( 1 + (-0.988 + 0.149i)T \) |
| 61 | \( 1 + (-0.826 - 0.563i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.900 + 0.433i)T \) |
| 73 | \( 1 + (0.733 - 0.680i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.955 + 0.294i)T \) |
| 89 | \( 1 + (0.955 + 0.294i)T \) |
| 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
−24.71752775749964159262625071978, −23.11989085749517350315148189146, −22.41009376480059547230382881453, −21.45406108349385423839633760127, −20.65434819035511671204478736589, −19.85986073145316409336691431468, −18.763259277645977341667026462074, −18.33754659024882431998606647328, −17.231538569439492167794265585527, −16.878822360562495482748965176521, −15.44913457351384213920905589489, −14.78743968599620860537550303946, −13.716355614565437873794980885775, −12.38956314587583300277718931059, −11.81642396463940663737940966681, −10.523333797456384489156157883868, −9.962364676602202188776864841863, −9.29457553294259304993863828585, −7.88404087597505512335669607396, −7.17808876357334596709319104934, −6.29586599961774559547199533933, −4.97039109484059955357624521118, −3.31728691517003431593576860491, −2.49045442914480526444636298528, −1.37368538365456508679777057131,
0.77603481607106365922832494603, 1.88997752128501366189024880380, 3.17329949957233176640929206405, 4.96283039934351247556440490126, 5.71559265550032712533976541380, 6.781997523032451669921559880870, 7.8260351934684831449915193720, 8.83687412819939607384548815713, 9.42190488802436933268007075256, 10.35376836618971513574079657440, 11.40852253560647694473108968867, 12.31842459095345160438009524312, 13.4906240875100077456196789822, 14.42414104495919002777935880262, 15.39998318720508096057267208200, 16.5382195730750381514251991427, 16.89692598823321894482064008512, 17.71969041081790445688180750199, 18.77086143745161587468636559786, 19.52588707865131708815400277736, 20.30621757144730847141891527179, 21.320915335680002525785773277924, 21.9025602878796581407964990066, 23.47635456755271471970658437254, 24.22652861639935581146968127872