L(s) = 1 | + (0.0747 + 0.997i)2-s + (−0.988 + 0.149i)4-s + (−0.222 − 0.974i)8-s + (0.900 + 0.433i)11-s + (0.0747 + 0.997i)13-s + (0.955 − 0.294i)16-s + (0.988 + 0.149i)17-s + (0.5 − 0.866i)19-s + (−0.365 + 0.930i)22-s + (0.623 + 0.781i)23-s + (−0.988 + 0.149i)26-s + (−0.365 − 0.930i)29-s + (0.5 − 0.866i)31-s + (0.365 + 0.930i)32-s + (−0.0747 + 0.997i)34-s + ⋯ |
L(s) = 1 | + (0.0747 + 0.997i)2-s + (−0.988 + 0.149i)4-s + (−0.222 − 0.974i)8-s + (0.900 + 0.433i)11-s + (0.0747 + 0.997i)13-s + (0.955 − 0.294i)16-s + (0.988 + 0.149i)17-s + (0.5 − 0.866i)19-s + (−0.365 + 0.930i)22-s + (0.623 + 0.781i)23-s + (−0.988 + 0.149i)26-s + (−0.365 − 0.930i)29-s + (0.5 − 0.866i)31-s + (0.365 + 0.930i)32-s + (−0.0747 + 0.997i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.395663220 + 1.010973692i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.395663220 + 1.010973692i\) |
\(L(1)\) |
\(\approx\) |
\(0.9998213431 + 0.5431100516i\) |
\(L(1)\) |
\(\approx\) |
\(0.9998213431 + 0.5431100516i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.0747 + 0.997i)T \) |
| 11 | \( 1 + (0.900 + 0.433i)T \) |
| 13 | \( 1 + (0.0747 + 0.997i)T \) |
| 17 | \( 1 + (0.988 + 0.149i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.623 + 0.781i)T \) |
| 29 | \( 1 + (-0.365 - 0.930i)T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + (-0.365 - 0.930i)T \) |
| 41 | \( 1 + (-0.733 + 0.680i)T \) |
| 43 | \( 1 + (0.733 + 0.680i)T \) |
| 47 | \( 1 + (-0.0747 - 0.997i)T \) |
| 53 | \( 1 + (0.365 - 0.930i)T \) |
| 59 | \( 1 + (-0.733 - 0.680i)T \) |
| 61 | \( 1 + (0.988 + 0.149i)T \) |
| 67 | \( 1 + (0.5 - 0.866i)T \) |
| 71 | \( 1 + (-0.623 - 0.781i)T \) |
| 73 | \( 1 + (0.826 + 0.563i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 + (-0.0747 + 0.997i)T \) |
| 89 | \( 1 + (0.0747 - 0.997i)T \) |
| 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
−19.635019349659437616281247697, −18.85036945592432933285186728056, −18.47544071020260651936800800205, −17.47177800181270790995087847228, −16.93619217186004683918331633688, −16.06753494441478164931594891479, −15.00167887006581989649851996567, −14.29461226233901944710771368827, −13.82391804663009370161671469535, −12.79603081702806181265468552932, −12.254572716672975944576442411932, −11.655159556630719229866603639603, −10.64375126306189767345635880319, −10.267374827649325352481919445479, −9.33150040745360652701154765124, −8.62302758503790257402367146353, −7.92428432308486092719395335976, −6.85586006780577828900582791223, −5.72999372053577477958611401086, −5.22179290039044458476277429947, −4.14602771850978218049406042062, −3.33459607943386919467021502991, −2.826293930327865485427236216831, −1.4672331172224865238477103963, −0.90901114174925813177528312592,
0.8096222150719405088726697640, 1.90740821976902021616004240979, 3.29859417162021997929049077325, 4.0385373643387936459189842683, 4.81301263974380950124434678408, 5.64701056360390008615886763192, 6.483956028728621322476978808286, 7.12350964854349882338133870241, 7.80790154481730257147379942919, 8.749769988807585513927117010825, 9.50244170407626878310674648001, 9.85769554172296542013347182846, 11.28275436938729861849520111212, 11.86017519195809620435941221788, 12.765711903706157297914840405572, 13.57919571970110137962530993211, 14.14808236450314650757623719533, 14.90306471933919562467549476150, 15.4620503279927281602776333779, 16.36931571386572589823989898945, 16.94248893563943293787743994859, 17.4951565940478996035111366225, 18.303096064301039800016780432923, 19.14343649254582296217669715690, 19.5559984739115490544934372080