L(s) = 1 | + (−0.974 + 0.222i)3-s + (0.900 − 0.433i)9-s + (0.900 + 0.433i)11-s + (0.433 − 0.900i)13-s + (−0.781 − 0.623i)17-s − 19-s + (−0.781 + 0.623i)23-s + (−0.781 + 0.623i)27-s + (−0.623 + 0.781i)29-s + 31-s + (−0.974 − 0.222i)33-s + (0.781 + 0.623i)37-s + (−0.222 + 0.974i)39-s + (0.222 + 0.974i)41-s + (−0.974 − 0.222i)43-s + ⋯ |
L(s) = 1 | + (−0.974 + 0.222i)3-s + (0.900 − 0.433i)9-s + (0.900 + 0.433i)11-s + (0.433 − 0.900i)13-s + (−0.781 − 0.623i)17-s − 19-s + (−0.781 + 0.623i)23-s + (−0.781 + 0.623i)27-s + (−0.623 + 0.781i)29-s + 31-s + (−0.974 − 0.222i)33-s + (0.781 + 0.623i)37-s + (−0.222 + 0.974i)39-s + (0.222 + 0.974i)41-s + (−0.974 − 0.222i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4426517512 + 0.6404287133i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4426517512 + 0.6404287133i\) |
\(L(1)\) |
\(\approx\) |
\(0.7492619455 + 0.07585450218i\) |
\(L(1)\) |
\(\approx\) |
\(0.7492619455 + 0.07585450218i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.974 + 0.222i)T \) |
| 11 | \( 1 + (0.900 + 0.433i)T \) |
| 13 | \( 1 + (0.433 - 0.900i)T \) |
| 17 | \( 1 + (-0.781 - 0.623i)T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + (-0.781 + 0.623i)T \) |
| 29 | \( 1 + (-0.623 + 0.781i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (0.781 + 0.623i)T \) |
| 41 | \( 1 + (0.222 + 0.974i)T \) |
| 43 | \( 1 + (-0.974 - 0.222i)T \) |
| 47 | \( 1 + (0.433 - 0.900i)T \) |
| 53 | \( 1 + (0.781 - 0.623i)T \) |
| 59 | \( 1 + (0.222 - 0.974i)T \) |
| 61 | \( 1 + (-0.623 + 0.781i)T \) |
| 67 | \( 1 - iT \) |
| 71 | \( 1 + (-0.623 - 0.781i)T \) |
| 73 | \( 1 + (-0.433 - 0.900i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (0.433 + 0.900i)T \) |
| 89 | \( 1 + (-0.900 + 0.433i)T \) |
| 97 | \( 1 - iT \) |
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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
−21.58443647334118679317540083276, −20.59143513134584519379307374903, −19.41180843076335160156476119257, −18.99048791139166977418000456898, −18.060022184449685082086004862516, −17.229541943454611626052387066073, −16.71742658876895553973313862691, −15.93184211585665350659194636853, −15.01923821729782002387969145295, −14.01118002890787422608762804467, −13.23754202783911874578679376671, −12.35968008575774808331140239906, −11.59111033534633991210518145752, −10.98490967583230853523656068015, −10.13520535874640758114217256457, −9.06976017484313235625033234940, −8.266880343889965739960438110341, −7.065515606127751198199671515384, −6.29046510819537237196094228615, −5.8594433879230248670188377961, −4.2904542781403841310891567781, −4.13232852977753497970109906152, −2.31916352440926575702423927733, −1.4035425275165965018864602745, −0.23445392218046184721713900037,
0.893744522147596266578184356749, 1.98401514540610373080196408263, 3.44242929550618837036112731230, 4.34456469725009653877512568509, 5.13556362338028828407995704857, 6.1907353602329252125476497840, 6.71237095037328376764511479919, 7.79652652169140087200268141975, 8.887696424862818513401092777137, 9.786974985778571182387597285099, 10.526248062612171642158764232785, 11.39678258322795266778419052512, 11.998401834945684022480100130309, 12.91731191660741726788162793096, 13.63792295009875240430918753926, 14.96170741005488920875614055373, 15.37368116855519037624601303447, 16.4078942814471565316395328225, 17.000252163630515305149392053501, 17.903625433996530236804278682855, 18.25741570400956658428591461070, 19.50306566569468626389459160564, 20.19639304350583151705543189584, 21.07277134496499805354680152830, 21.99355280082554708072785798788