L(s) = 1 | + (0.487 + 0.873i)2-s + (0.633 − 0.773i)3-s + (−0.525 + 0.850i)4-s + (−0.367 + 0.930i)5-s + (0.984 + 0.176i)6-s + (−0.110 − 0.993i)7-s + (−0.999 − 0.0442i)8-s + (−0.197 − 0.980i)9-s + (−0.991 + 0.132i)10-s + (0.699 − 0.714i)11-s + (0.325 + 0.945i)12-s + (−0.598 − 0.801i)13-s + (0.814 − 0.580i)14-s + (0.487 + 0.873i)15-s + (−0.448 − 0.894i)16-s + (0.154 − 0.988i)17-s + ⋯ |
L(s) = 1 | + (0.487 + 0.873i)2-s + (0.633 − 0.773i)3-s + (−0.525 + 0.850i)4-s + (−0.367 + 0.930i)5-s + (0.984 + 0.176i)6-s + (−0.110 − 0.993i)7-s + (−0.999 − 0.0442i)8-s + (−0.197 − 0.980i)9-s + (−0.991 + 0.132i)10-s + (0.699 − 0.714i)11-s + (0.325 + 0.945i)12-s + (−0.598 − 0.801i)13-s + (0.814 − 0.580i)14-s + (0.487 + 0.873i)15-s + (−0.448 − 0.894i)16-s + (0.154 − 0.988i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.415330866 - 0.5441150808i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.415330866 - 0.5441150808i\) |
\(L(1)\) |
\(\approx\) |
\(1.295144740 + 0.06926050262i\) |
\(L(1)\) |
\(\approx\) |
\(1.295144740 + 0.06926050262i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 569 | \( 1 \) |
good | 2 | \( 1 + (0.487 + 0.873i)T \) |
| 3 | \( 1 + (0.633 - 0.773i)T \) |
| 5 | \( 1 + (-0.367 + 0.930i)T \) |
| 7 | \( 1 + (-0.110 - 0.993i)T \) |
| 11 | \( 1 + (0.699 - 0.714i)T \) |
| 13 | \( 1 + (-0.598 - 0.801i)T \) |
| 17 | \( 1 + (0.154 - 0.988i)T \) |
| 19 | \( 1 + (-0.525 - 0.850i)T \) |
| 23 | \( 1 + (-0.839 + 0.544i)T \) |
| 29 | \( 1 + (0.154 + 0.988i)T \) |
| 31 | \( 1 + (0.996 - 0.0883i)T \) |
| 37 | \( 1 + (-0.110 - 0.993i)T \) |
| 41 | \( 1 + (-0.839 + 0.544i)T \) |
| 43 | \( 1 + (0.487 - 0.873i)T \) |
| 47 | \( 1 + (0.759 + 0.650i)T \) |
| 53 | \( 1 + (0.984 + 0.176i)T \) |
| 59 | \( 1 + (0.759 + 0.650i)T \) |
| 61 | \( 1 + (0.903 - 0.428i)T \) |
| 67 | \( 1 + (0.154 + 0.988i)T \) |
| 71 | \( 1 + (-0.999 + 0.0442i)T \) |
| 73 | \( 1 + (-0.525 - 0.850i)T \) |
| 79 | \( 1 + (0.937 - 0.346i)T \) |
| 83 | \( 1 + (-0.975 + 0.219i)T \) |
| 89 | \( 1 + (0.996 + 0.0883i)T \) |
| 97 | \( 1 + (-0.730 - 0.683i)T \) |
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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
−23.11529989452507196383933889889, −22.23769348200890735734502992757, −21.56419266819344809985533273228, −20.89374712397440513106970652847, −20.19033055929889211753038088172, −19.29439243265663068965360860773, −18.98255861698308713415017332286, −17.4298906963585174951500527156, −16.53384970923668924605569693074, −15.441077018862614350065796176415, −14.896602910960844748440115146627, −14.09495116556494855053375454198, −13.00230844715675074227562918460, −12.08935746774148127918423896343, −11.7432888737655879335898814301, −10.21480571016355686050866451797, −9.67930336934299198114166557698, −8.76820727836684766811891836272, −8.19016850625820944480731045183, −6.31537614155898643631593260758, −5.244256878907641544185143633196, −4.3227696337981094511084309363, −3.86033262780642218276354629221, −2.415278915669250385505892086029, −1.70075014052624189316860941999,
0.60713042267936538658439040057, 2.632230600460147358501931320753, 3.38254310143090681099065979483, 4.235351090990439864519206817442, 5.7328715068540874359704731394, 6.80459512787524849754838078163, 7.20066961367079726334490932276, 7.98929339837672833600770039952, 8.967854419681508720369679238229, 10.13054999357688558490035012268, 11.424773312710785678765841057007, 12.242839414350954686595278728493, 13.36003037108400710902974241361, 13.94034710719643334293210063814, 14.52629740403540802811464631462, 15.37438771158279081940416531335, 16.30648919002999106987933586047, 17.47049375568624488991744996694, 17.9019691026634389723565375771, 19.03729291215332117762318474788, 19.70810989854167863489694539560, 20.57924835840201382233181497364, 21.85126343155726995401687041880, 22.50631071499833753356764290716, 23.39744540959036758255808271847