L(s) = 1 | + (0.354 + 0.935i)2-s + (−0.996 − 0.0804i)3-s + (−0.748 + 0.663i)4-s + (−0.799 + 0.600i)5-s + (−0.278 − 0.960i)6-s + (−0.885 − 0.464i)8-s + (0.987 + 0.160i)9-s + (−0.845 − 0.534i)10-s + (−0.987 + 0.160i)11-s + (0.799 − 0.600i)12-s + (0.845 − 0.534i)15-s + (0.120 − 0.992i)16-s + (0.885 + 0.464i)17-s + (0.200 + 0.979i)18-s + (0.5 − 0.866i)19-s + (0.200 − 0.979i)20-s + ⋯ |
L(s) = 1 | + (0.354 + 0.935i)2-s + (−0.996 − 0.0804i)3-s + (−0.748 + 0.663i)4-s + (−0.799 + 0.600i)5-s + (−0.278 − 0.960i)6-s + (−0.885 − 0.464i)8-s + (0.987 + 0.160i)9-s + (−0.845 − 0.534i)10-s + (−0.987 + 0.160i)11-s + (0.799 − 0.600i)12-s + (0.845 − 0.534i)15-s + (0.120 − 0.992i)16-s + (0.885 + 0.464i)17-s + (0.200 + 0.979i)18-s + (0.5 − 0.866i)19-s + (0.200 − 0.979i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3389376875 + 0.7671933151i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3389376875 + 0.7671933151i\) |
\(L(1)\) |
\(\approx\) |
\(0.5918291438 + 0.4494683786i\) |
\(L(1)\) |
\(\approx\) |
\(0.5918291438 + 0.4494683786i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.354 + 0.935i)T \) |
| 3 | \( 1 + (-0.996 - 0.0804i)T \) |
| 5 | \( 1 + (-0.799 + 0.600i)T \) |
| 11 | \( 1 + (-0.987 + 0.160i)T \) |
| 17 | \( 1 + (0.885 + 0.464i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (0.987 + 0.160i)T \) |
| 31 | \( 1 + (-0.692 + 0.721i)T \) |
| 37 | \( 1 + (0.970 + 0.239i)T \) |
| 41 | \( 1 + (-0.428 - 0.903i)T \) |
| 43 | \( 1 + (0.692 + 0.721i)T \) |
| 47 | \( 1 + (0.200 - 0.979i)T \) |
| 53 | \( 1 + (-0.845 + 0.534i)T \) |
| 59 | \( 1 + (-0.120 - 0.992i)T \) |
| 61 | \( 1 + (-0.0402 + 0.999i)T \) |
| 67 | \( 1 + (0.200 - 0.979i)T \) |
| 71 | \( 1 + (0.996 + 0.0804i)T \) |
| 73 | \( 1 + (0.632 + 0.774i)T \) |
| 79 | \( 1 + (-0.200 + 0.979i)T \) |
| 83 | \( 1 + (-0.568 - 0.822i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (0.919 - 0.391i)T \) |
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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
−20.88127253674062181392843124849, −20.47432859867193707601179179090, −19.33653171217779898668017599896, −18.68658386704999297863632487317, −18.139426136812901325642158949458, −17.07275475750544018503683583782, −16.31181870023798386125497868378, −15.61391029689212226981952274209, −14.74445008022896301143712705568, −13.63899868555207500323323657263, −12.72022870134450467922776809113, −12.38313895196035199284947848084, −11.47619026909727960425696205559, −10.99072337384046891318791704300, −10.05659528224861388685412327835, −9.37414862259702303703787255628, −8.17019189471552342261702864826, −7.40570017681170501206723578829, −6.02214089959434310210272802715, −5.27241278223287783115502233837, −4.68982300230237298248243502799, −3.76721435537743478957153329422, −2.83904811829433428119978074044, −1.37548079577481394174361134118, −0.550551293341690944237104891769,
0.81635505189813695369696519106, 2.75518858464302156402124995450, 3.666467867653825060030633870245, 4.73754590852880718865240779303, 5.27169545105071774103541165481, 6.26367863598138659729904193431, 7.08646219104220217014273770681, 7.57600974670790521961695414404, 8.46202776402148875237608899111, 9.7024965761979503597141376003, 10.62353777577512946555059238210, 11.35100550359820413719445422226, 12.33161203339528805618679503270, 12.79418245856812736018454340358, 13.79893042338367269279023953941, 14.74087736911213622075699705285, 15.55616211960694282427436824953, 15.92230845834327163949320103341, 16.80159437342231657377299522218, 17.54977800454932480464241596222, 18.3736749272007236749042887952, 18.7474156728198281201963980021, 19.85871369903465152801174749897, 21.189956526241453195387912823968, 21.6802342691152488271434107347