L(s) = 1 | + (0.856 − 0.515i)2-s + (−0.370 − 0.928i)3-s + (0.468 − 0.883i)4-s + (−0.994 + 0.108i)5-s + (−0.796 − 0.605i)6-s + (−0.947 + 0.319i)7-s + (−0.0541 − 0.998i)8-s + (−0.725 + 0.687i)9-s + (−0.796 + 0.605i)10-s + (0.561 − 0.827i)11-s + (−0.994 − 0.108i)12-s + (0.725 + 0.687i)13-s + (−0.647 + 0.762i)14-s + (0.468 + 0.883i)15-s + (−0.561 − 0.827i)16-s + (−0.947 − 0.319i)17-s + ⋯ |
L(s) = 1 | + (0.856 − 0.515i)2-s + (−0.370 − 0.928i)3-s + (0.468 − 0.883i)4-s + (−0.994 + 0.108i)5-s + (−0.796 − 0.605i)6-s + (−0.947 + 0.319i)7-s + (−0.0541 − 0.998i)8-s + (−0.725 + 0.687i)9-s + (−0.796 + 0.605i)10-s + (0.561 − 0.827i)11-s + (−0.994 − 0.108i)12-s + (0.725 + 0.687i)13-s + (−0.647 + 0.762i)14-s + (0.468 + 0.883i)15-s + (−0.561 − 0.827i)16-s + (−0.947 − 0.319i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 + 0.0870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 + 0.0870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.05320709956 - 1.220016541i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.05320709956 - 1.220016541i\) |
\(L(1)\) |
\(\approx\) |
\(0.7555154866 - 0.7764156431i\) |
\(L(1)\) |
\(\approx\) |
\(0.7555154866 - 0.7764156431i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 59 | \( 1 \) |
good | 2 | \( 1 + (0.856 - 0.515i)T \) |
| 3 | \( 1 + (-0.370 - 0.928i)T \) |
| 5 | \( 1 + (-0.994 + 0.108i)T \) |
| 7 | \( 1 + (-0.947 + 0.319i)T \) |
| 11 | \( 1 + (0.561 - 0.827i)T \) |
| 13 | \( 1 + (0.725 + 0.687i)T \) |
| 17 | \( 1 + (-0.947 - 0.319i)T \) |
| 19 | \( 1 + (-0.161 - 0.986i)T \) |
| 23 | \( 1 + (-0.267 - 0.963i)T \) |
| 29 | \( 1 + (-0.856 - 0.515i)T \) |
| 31 | \( 1 + (0.161 - 0.986i)T \) |
| 37 | \( 1 + (-0.0541 + 0.998i)T \) |
| 41 | \( 1 + (0.267 - 0.963i)T \) |
| 43 | \( 1 + (0.561 + 0.827i)T \) |
| 47 | \( 1 + (0.994 + 0.108i)T \) |
| 53 | \( 1 + (0.796 + 0.605i)T \) |
| 61 | \( 1 + (0.856 - 0.515i)T \) |
| 67 | \( 1 + (-0.0541 - 0.998i)T \) |
| 71 | \( 1 + (-0.994 - 0.108i)T \) |
| 73 | \( 1 + (-0.647 + 0.762i)T \) |
| 79 | \( 1 + (-0.370 + 0.928i)T \) |
| 83 | \( 1 + (-0.907 + 0.419i)T \) |
| 89 | \( 1 + (0.856 + 0.515i)T \) |
| 97 | \( 1 + (-0.647 - 0.762i)T \) |
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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
−32.96546656215547614267698171710, −32.075052739297602377726066391, −31.167230037125328518859246269252, −29.89985823332285735982812886338, −28.49254014422706602711346664762, −27.3143807036446479292410251112, −26.236056989673847580304402117596, −25.21231282544451125895734500429, −23.511271814040815886971415289832, −22.893291161398208481487308176716, −22.09161730219262839904107351480, −20.572319069688793351195199361365, −19.78761137910361074296657399809, −17.55757760942753948367372303114, −16.31809751938766247648976668767, −15.6563344680523095775909404204, −14.67742583569185161686544856233, −12.950754050843690035258997964524, −11.86572061501920559888324104853, −10.58106278192372097161041282213, −8.82505254370860459483493711014, −7.17738505888766371641094935769, −5.81014533173642734991821490061, −4.1988933892459122396153269300, −3.49266788821337171240072202580,
0.51695089609634023128322605945, 2.627549058471976591303923626475, 4.14423850013148393872833159822, 6.072459917101450137059585682677, 6.92808787012669584953447206826, 8.87488366561585410835558632599, 11.04508254837863150227416336660, 11.72713263090613639951072579677, 12.90583674973779254465738328079, 13.862233448702887393551496091135, 15.46397614268248362872031023081, 16.53559360263013156464829934214, 18.667507327033069889357964937539, 19.209576902026307505039289032798, 20.23581311165696318984103504669, 22.106350613430951995206998194694, 22.71918757729666956194394979051, 23.90074987209572320452522793970, 24.54849020485858332154269843084, 26.14683564290855041172520707962, 27.944992363413005563612052066803, 28.75298148092398479219446509477, 29.81182470211600454315379267989, 30.77263331368107093358198135942, 31.59208664281292257535904740486