L(s) = 1 | + (−0.453 + 0.891i)3-s + 7-s + (−0.587 − 0.809i)9-s + (−0.156 − 0.987i)11-s + (0.987 + 0.156i)13-s + (0.951 + 0.309i)17-s + (−0.453 − 0.891i)19-s + (−0.453 + 0.891i)21-s + (−0.809 − 0.587i)23-s + (0.987 − 0.156i)27-s + (−0.891 − 0.453i)29-s + (0.309 − 0.951i)31-s + (0.951 + 0.309i)33-s + (0.156 − 0.987i)37-s + (−0.587 + 0.809i)39-s + ⋯ |
L(s) = 1 | + (−0.453 + 0.891i)3-s + 7-s + (−0.587 − 0.809i)9-s + (−0.156 − 0.987i)11-s + (0.987 + 0.156i)13-s + (0.951 + 0.309i)17-s + (−0.453 − 0.891i)19-s + (−0.453 + 0.891i)21-s + (−0.809 − 0.587i)23-s + (0.987 − 0.156i)27-s + (−0.891 − 0.453i)29-s + (0.309 − 0.951i)31-s + (0.951 + 0.309i)33-s + (0.156 − 0.987i)37-s + (−0.587 + 0.809i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5012465172 - 0.7072812681i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5012465172 - 0.7072812681i\) |
\(L(1)\) |
\(\approx\) |
\(0.9108385591 + 0.05719404173i\) |
\(L(1)\) |
\(\approx\) |
\(0.9108385591 + 0.05719404173i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.453 + 0.891i)T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + (-0.156 - 0.987i)T \) |
| 13 | \( 1 + (0.987 + 0.156i)T \) |
| 17 | \( 1 + (0.951 + 0.309i)T \) |
| 19 | \( 1 + (-0.453 - 0.891i)T \) |
| 23 | \( 1 + (-0.809 - 0.587i)T \) |
| 29 | \( 1 + (-0.891 - 0.453i)T \) |
| 31 | \( 1 + (0.309 - 0.951i)T \) |
| 37 | \( 1 + (0.156 - 0.987i)T \) |
| 41 | \( 1 + (0.587 + 0.809i)T \) |
| 43 | \( 1 + (-0.707 + 0.707i)T \) |
| 47 | \( 1 + (-0.951 + 0.309i)T \) |
| 53 | \( 1 + (-0.891 - 0.453i)T \) |
| 59 | \( 1 + (-0.987 - 0.156i)T \) |
| 61 | \( 1 + (-0.987 + 0.156i)T \) |
| 67 | \( 1 + (-0.891 + 0.453i)T \) |
| 71 | \( 1 + (-0.951 + 0.309i)T \) |
| 73 | \( 1 + (-0.809 - 0.587i)T \) |
| 79 | \( 1 + (0.309 + 0.951i)T \) |
| 83 | \( 1 + (-0.891 + 0.453i)T \) |
| 89 | \( 1 + (-0.587 + 0.809i)T \) |
| 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
−22.54754586277045082358048528593, −21.44545329089473122035916985419, −20.64396556244202150166534334715, −19.99453090580434616825427089644, −18.77829227147491076338926803627, −18.32653300844950659152938038059, −17.604649731931485016703693994559, −16.88741421883156010958874598137, −15.941219223659816309962964437794, −14.862081920927036778811957822523, −14.115779820094467431086843840328, −13.3283458073925660239256943768, −12.31146691410359832573918998825, −11.84824755682035890183369876673, −10.858931718434913685477252467988, −10.12829617551836292122561363503, −8.76507973922200777376215962127, −7.88628252796592682606200530039, −7.381562112432614679130887640520, −6.21137883679708955411039722985, −5.444357137492493347377547265, −4.53893228597987205787388448858, −3.24784295355413052664732105988, −1.77585937282117850275237608216, −1.40621860291298101492467851149,
0.204854458893208948390170624221, 1.398224339583099140442552495553, 2.87053574436634245705274371348, 3.92769202293263607587266932562, 4.65110765663722756076012177823, 5.7514611505349440333586600369, 6.22954215914019453388298505031, 7.813254731417123063247295736989, 8.49404116829390949505371096132, 9.38281271109282391268052905595, 10.4162753910576644686311978283, 11.22397926269091471340599638565, 11.51956431962238577384660927790, 12.80032839240049944254979996160, 13.85346733061549478359995804891, 14.62167299588287043993546421318, 15.35425291542507199803293780831, 16.311831851168090302600579183880, 16.793994711193178247043382164061, 17.79915382952643529155506324932, 18.4264775193667838450853560466, 19.45841151359025526998463581359, 20.56909503089147935560756382329, 21.131151621088075292873489276065, 21.62585071230769134080247702466