L(s) = 1 | + (0.587 − 0.809i)2-s + (−0.951 + 0.309i)3-s + (−0.309 − 0.951i)4-s + (−0.309 + 0.951i)6-s + (−0.951 − 0.309i)8-s + (0.809 − 0.587i)9-s + (−0.809 − 0.587i)11-s + (0.587 + 0.809i)12-s + (−0.587 − 0.809i)13-s + (−0.809 + 0.587i)16-s + (−0.951 − 0.309i)17-s − i·18-s + (0.309 − 0.951i)19-s + (−0.951 + 0.309i)22-s + (−0.587 + 0.809i)23-s + 24-s + ⋯ |
L(s) = 1 | + (0.587 − 0.809i)2-s + (−0.951 + 0.309i)3-s + (−0.309 − 0.951i)4-s + (−0.309 + 0.951i)6-s + (−0.951 − 0.309i)8-s + (0.809 − 0.587i)9-s + (−0.809 − 0.587i)11-s + (0.587 + 0.809i)12-s + (−0.587 − 0.809i)13-s + (−0.809 + 0.587i)16-s + (−0.951 − 0.309i)17-s − i·18-s + (0.309 − 0.951i)19-s + (−0.951 + 0.309i)22-s + (−0.587 + 0.809i)23-s + 24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.08477136485 - 0.6710344490i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08477136485 - 0.6710344490i\) |
\(L(1)\) |
\(\approx\) |
\(0.6454144820 - 0.4985570024i\) |
\(L(1)\) |
\(\approx\) |
\(0.6454144820 - 0.4985570024i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.587 - 0.809i)T \) |
| 3 | \( 1 + (-0.951 + 0.309i)T \) |
| 11 | \( 1 + (-0.809 - 0.587i)T \) |
| 13 | \( 1 + (-0.587 - 0.809i)T \) |
| 17 | \( 1 + (-0.951 - 0.309i)T \) |
| 19 | \( 1 + (0.309 - 0.951i)T \) |
| 23 | \( 1 + (-0.587 + 0.809i)T \) |
| 29 | \( 1 + (-0.309 - 0.951i)T \) |
| 31 | \( 1 + (-0.309 + 0.951i)T \) |
| 37 | \( 1 + (-0.587 - 0.809i)T \) |
| 41 | \( 1 + (0.809 - 0.587i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (0.951 - 0.309i)T \) |
| 53 | \( 1 + (0.951 - 0.309i)T \) |
| 59 | \( 1 + (-0.809 + 0.587i)T \) |
| 61 | \( 1 + (0.809 + 0.587i)T \) |
| 67 | \( 1 + (0.951 + 0.309i)T \) |
| 71 | \( 1 + (0.309 + 0.951i)T \) |
| 73 | \( 1 + (0.587 - 0.809i)T \) |
| 79 | \( 1 + (-0.309 - 0.951i)T \) |
| 83 | \( 1 + (0.951 + 0.309i)T \) |
| 89 | \( 1 + (-0.809 - 0.587i)T \) |
| 97 | \( 1 + (0.951 - 0.309i)T \) |
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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
−27.88023888018711241353936646522, −26.72751246162164942451393198211, −25.92101762466914044309792273205, −24.602464762542667623165790123337, −24.07326228727313998682381747063, −23.1480413455025342899454462875, −22.33796725990539454142810113111, −21.555586166036199739312061557563, −20.37398083541901551512737998770, −18.68734968302945561977194429274, −17.94395158145142340426022430976, −16.93002336404092373429624415604, −16.19993343133172058074289698935, −15.17354187837885087346271505331, −14.009663007847245641766911360485, −12.85374318623664427795403811062, −12.22199526090850785969483567217, −11.037458736707192556775252000665, −9.66838948904297265560396316810, −8.0849017288814969117519927380, −7.10648357005713286255823241680, −6.18281271511041836773313272724, −5.06365447251871176906948980846, −4.170390527460566014878737701527, −2.21068031232296435927935911146,
0.485364613186329620367917158646, 2.4124122595640504122034548594, 3.821871181525172990974351720481, 5.08700391749387491784801041367, 5.73795384253506985204142789745, 7.18302300939542769906096534484, 9.0507486434013061925467095747, 10.21818591936789310495100451279, 10.95694406356040468447907897163, 11.87380584408438439706255620754, 12.89302450375275380026194175664, 13.77825505210041061958865249319, 15.32591702739054573793987177237, 15.86284960914257831859425899602, 17.48962791740799344763494627928, 18.14387507633122586054707966847, 19.36793923823372489834998836571, 20.39372285217732093598191626594, 21.449653296790893093590165481338, 22.10503816896232570437319393383, 22.94066845198646899379469792697, 23.89421275410479292388262075833, 24.593604041312833411242715240589, 26.43123089499340428939131510617, 27.26593334597279393042113568333