L(s) = 1 | + (0.449 − 0.893i)2-s + (−0.595 − 0.803i)4-s + (−0.999 + 0.0380i)5-s + (−0.123 + 0.992i)7-s + (−0.985 + 0.170i)8-s + (−0.415 + 0.909i)10-s + (−0.00951 − 0.999i)13-s + (0.830 + 0.556i)14-s + (−0.290 + 0.956i)16-s + (0.941 + 0.336i)17-s + (0.0285 + 0.999i)19-s + (0.625 + 0.780i)20-s + (−0.981 − 0.189i)23-s + (0.997 − 0.0760i)25-s + (−0.897 − 0.441i)26-s + ⋯ |
L(s) = 1 | + (0.449 − 0.893i)2-s + (−0.595 − 0.803i)4-s + (−0.999 + 0.0380i)5-s + (−0.123 + 0.992i)7-s + (−0.985 + 0.170i)8-s + (−0.415 + 0.909i)10-s + (−0.00951 − 0.999i)13-s + (0.830 + 0.556i)14-s + (−0.290 + 0.956i)16-s + (0.941 + 0.336i)17-s + (0.0285 + 0.999i)19-s + (0.625 + 0.780i)20-s + (−0.981 − 0.189i)23-s + (0.997 − 0.0760i)25-s + (−0.897 − 0.441i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.576 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.576 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5028584070 - 0.9705918796i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5028584070 - 0.9705918796i\) |
\(L(1)\) |
\(\approx\) |
\(0.8375079886 - 0.4889181345i\) |
\(L(1)\) |
\(\approx\) |
\(0.8375079886 - 0.4889181345i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.449 - 0.893i)T \) |
| 5 | \( 1 + (-0.999 + 0.0380i)T \) |
| 7 | \( 1 + (-0.123 + 0.992i)T \) |
| 13 | \( 1 + (-0.00951 - 0.999i)T \) |
| 17 | \( 1 + (0.941 + 0.336i)T \) |
| 19 | \( 1 + (0.0285 + 0.999i)T \) |
| 23 | \( 1 + (-0.981 - 0.189i)T \) |
| 29 | \( 1 + (-0.432 - 0.901i)T \) |
| 31 | \( 1 + (0.749 - 0.662i)T \) |
| 37 | \( 1 + (-0.736 - 0.676i)T \) |
| 41 | \( 1 + (0.964 - 0.263i)T \) |
| 43 | \( 1 + (0.786 + 0.618i)T \) |
| 47 | \( 1 + (0.935 - 0.353i)T \) |
| 53 | \( 1 + (-0.974 - 0.226i)T \) |
| 59 | \( 1 + (0.710 - 0.703i)T \) |
| 61 | \( 1 + (-0.548 + 0.836i)T \) |
| 67 | \( 1 + (0.0475 - 0.998i)T \) |
| 71 | \( 1 + (-0.610 - 0.791i)T \) |
| 73 | \( 1 + (-0.516 - 0.856i)T \) |
| 79 | \( 1 + (0.969 - 0.244i)T \) |
| 83 | \( 1 + (-0.905 - 0.424i)T \) |
| 89 | \( 1 + (0.959 - 0.281i)T \) |
| 97 | \( 1 + (0.999 + 0.0380i)T \) |
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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
−21.94598781704239354262569131684, −20.96918699874078377444240518656, −20.22566416776035376813811778772, −19.31686323311255707630387545053, −18.61422533775022838538380144772, −17.53703781660043754732157547786, −16.88165941621545434251582873286, −16.0170091980694698207703438090, −15.76621781187448742169899236023, −14.48151242582930121321994003827, −14.120898102645389557426663697682, −13.21619237238256169848776105748, −12.27550559135880217937809185165, −11.65821606385320827024115880421, −10.65866732086247081629441531821, −9.51435783761190642760026239307, −8.6669891775462964416264739033, −7.71625899864828545941731620057, −7.1820098100732391082283562513, −6.49124847472868752914822591769, −5.23780562841944752348868851314, −4.37934121574304986011640038485, −3.802860218599065946153313458009, −2.86973136251669167037184046607, −0.99734948367327140898852402197,
0.48849963710553555689586203424, 1.87515835761290961965799389972, 2.897675005373461254702200141599, 3.63370993544346015762823485936, 4.47061610257909871033560303946, 5.64235097294179569382310450557, 6.058616622398499564207052117664, 7.72433194870339234066417949165, 8.26152536800959042001122018750, 9.31754973257295357931446926982, 10.17340308788166475597866396199, 10.92901521156868995492055188218, 11.98216084284335469957968558366, 12.225998730777543186674534619978, 12.9798788822418530076524896672, 14.15227278682909440919026930568, 14.86931249626464741353943409707, 15.4945372477857306894623965431, 16.25617035609217012443405331481, 17.5463543400634588762238962231, 18.38730754354733630320746394685, 19.086089309575411261433445291298, 19.4799707469417153006502858171, 20.59011969727699436912129983351, 20.95250856621527568406265426212