L(s) = 1 | + (−0.557 − 0.830i)2-s + (−0.377 + 0.925i)4-s + (−0.591 − 0.806i)5-s + (0.979 − 0.202i)8-s + (−0.339 + 0.940i)10-s + (0.301 + 0.953i)11-s + (0.992 + 0.122i)13-s + (−0.714 − 0.699i)16-s + (−0.377 − 0.925i)17-s + (0.959 − 0.281i)19-s + (0.970 − 0.242i)20-s + (0.623 − 0.781i)22-s + (−0.301 + 0.953i)25-s + (−0.452 − 0.891i)26-s + (0.377 + 0.925i)29-s + ⋯ |
L(s) = 1 | + (−0.557 − 0.830i)2-s + (−0.377 + 0.925i)4-s + (−0.591 − 0.806i)5-s + (0.979 − 0.202i)8-s + (−0.339 + 0.940i)10-s + (0.301 + 0.953i)11-s + (0.992 + 0.122i)13-s + (−0.714 − 0.699i)16-s + (−0.377 − 0.925i)17-s + (0.959 − 0.281i)19-s + (0.970 − 0.242i)20-s + (0.623 − 0.781i)22-s + (−0.301 + 0.953i)25-s + (−0.452 − 0.891i)26-s + (0.377 + 0.925i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.508 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.508 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4889515711 - 0.8565879324i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4889515711 - 0.8565879324i\) |
\(L(1)\) |
\(\approx\) |
\(0.6599374429 - 0.3615561184i\) |
\(L(1)\) |
\(\approx\) |
\(0.6599374429 - 0.3615561184i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + (-0.557 - 0.830i)T \) |
| 5 | \( 1 + (-0.591 - 0.806i)T \) |
| 11 | \( 1 + (0.301 + 0.953i)T \) |
| 13 | \( 1 + (0.992 + 0.122i)T \) |
| 17 | \( 1 + (-0.377 - 0.925i)T \) |
| 19 | \( 1 + (0.959 - 0.281i)T \) |
| 29 | \( 1 + (0.377 + 0.925i)T \) |
| 31 | \( 1 + (-0.415 - 0.909i)T \) |
| 37 | \( 1 + (0.182 - 0.983i)T \) |
| 41 | \( 1 + (-0.591 - 0.806i)T \) |
| 43 | \( 1 + (-0.979 - 0.202i)T \) |
| 47 | \( 1 + (-0.900 + 0.433i)T \) |
| 53 | \( 1 + (-0.101 + 0.994i)T \) |
| 59 | \( 1 + (0.339 - 0.940i)T \) |
| 61 | \( 1 + (0.452 - 0.891i)T \) |
| 67 | \( 1 + (-0.142 - 0.989i)T \) |
| 71 | \( 1 + (0.862 + 0.505i)T \) |
| 73 | \( 1 + (-0.986 - 0.162i)T \) |
| 79 | \( 1 + (0.841 + 0.540i)T \) |
| 83 | \( 1 + (0.262 + 0.965i)T \) |
| 89 | \( 1 + (-0.768 + 0.639i)T \) |
| 97 | \( 1 + (0.654 - 0.755i)T \) |
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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
−18.94847060747488519063819136265, −18.22811381166144536971423355509, −17.874640898992554563242080338707, −16.846351352222352830127116235128, −16.23623677610202889857111382366, −15.70224983646238430958116032044, −14.97863897697845223424656669972, −14.43309632238907226391915624310, −13.62991319346322319583492862535, −13.13629695878259811768538231668, −11.62748734407042357100136725392, −11.42563268517699755870740392112, −10.39506678388817546812486404542, −10.05491164446036759138633029170, −8.851695264114521750654074677970, −8.35111906734619651655243465689, −7.8035443476757498177881230914, −6.81386502450317579308913164054, −6.33493783805408666595923884488, −5.69347119126130489207820901900, −4.66218823087122064411443351687, −3.706841481415918103465499413651, −3.13259608710726130463357072111, −1.76097432422699627999060552784, −0.874188996702209711539611259117,
0.48003359185529612233714552109, 1.3682766666287092100907351039, 2.121243808563377596193064935347, 3.26470361721634931528795641000, 3.8684944368417900117346452958, 4.69930819639512938529948679924, 5.27945326092575587306169426836, 6.688496479235764038334566721053, 7.39309674455826095814362235455, 8.055614528879283260858844494385, 8.9257671833540341140018775532, 9.31026771188214066333226177003, 10.038663646977072119386977747001, 11.1393545121764218894595223375, 11.44988802735691246519559462902, 12.26822312613979474490374288110, 12.77908147474042272221952205823, 13.53114375097433018642750877952, 14.2136549858414465725503203706, 15.43618759854996525862037162734, 15.93863887329357982264587681088, 16.60903162934820804736017446560, 17.28802589493304690690320413690, 18.15539172518047002113212313861, 18.43085828076445659253583670484