L(s) = 1 | + (−0.320 − 0.947i)2-s + (0.674 − 0.737i)3-s + (−0.794 + 0.606i)4-s + (0.375 − 0.926i)5-s + (−0.915 − 0.403i)6-s + (0.992 + 0.118i)7-s + (0.829 + 0.558i)8-s + (−0.0887 − 0.996i)9-s + (−0.998 − 0.0592i)10-s + (−0.630 + 0.776i)11-s + (−0.0887 + 0.996i)12-s + (0.147 − 0.989i)13-s + (−0.205 − 0.978i)14-s + (−0.430 − 0.902i)15-s + (0.263 − 0.964i)16-s + (0.0296 + 0.999i)17-s + ⋯ |
L(s) = 1 | + (−0.320 − 0.947i)2-s + (0.674 − 0.737i)3-s + (−0.794 + 0.606i)4-s + (0.375 − 0.926i)5-s + (−0.915 − 0.403i)6-s + (0.992 + 0.118i)7-s + (0.829 + 0.558i)8-s + (−0.0887 − 0.996i)9-s + (−0.998 − 0.0592i)10-s + (−0.630 + 0.776i)11-s + (−0.0887 + 0.996i)12-s + (0.147 − 0.989i)13-s + (−0.205 − 0.978i)14-s + (−0.430 − 0.902i)15-s + (0.263 − 0.964i)16-s + (0.0296 + 0.999i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.545 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.545 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5257477516 - 0.9700843929i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5257477516 - 0.9700843929i\) |
\(L(1)\) |
\(\approx\) |
\(0.8040394657 - 0.7560004586i\) |
\(L(1)\) |
\(\approx\) |
\(0.8040394657 - 0.7560004586i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 107 | \( 1 \) |
good | 2 | \( 1 + (-0.320 - 0.947i)T \) |
| 3 | \( 1 + (0.674 - 0.737i)T \) |
| 5 | \( 1 + (0.375 - 0.926i)T \) |
| 7 | \( 1 + (0.992 + 0.118i)T \) |
| 11 | \( 1 + (-0.630 + 0.776i)T \) |
| 13 | \( 1 + (0.147 - 0.989i)T \) |
| 17 | \( 1 + (0.0296 + 0.999i)T \) |
| 19 | \( 1 + (-0.956 + 0.292i)T \) |
| 23 | \( 1 + (-0.205 + 0.978i)T \) |
| 29 | \( 1 + (-0.533 + 0.845i)T \) |
| 31 | \( 1 + (0.582 - 0.812i)T \) |
| 37 | \( 1 + (-0.984 - 0.176i)T \) |
| 41 | \( 1 + (0.889 - 0.456i)T \) |
| 43 | \( 1 + (0.375 + 0.926i)T \) |
| 47 | \( 1 + (0.889 + 0.456i)T \) |
| 53 | \( 1 + (-0.320 + 0.947i)T \) |
| 59 | \( 1 + (-0.533 - 0.845i)T \) |
| 61 | \( 1 + (0.992 - 0.118i)T \) |
| 67 | \( 1 + (0.829 - 0.558i)T \) |
| 71 | \( 1 + (0.674 + 0.737i)T \) |
| 73 | \( 1 + (0.937 - 0.348i)T \) |
| 79 | \( 1 + (-0.861 + 0.508i)T \) |
| 83 | \( 1 + (0.972 - 0.234i)T \) |
| 89 | \( 1 + (-0.915 + 0.403i)T \) |
| 97 | \( 1 + (-0.998 - 0.0592i)T \) |
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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
−30.22106756305205626844085294057, −28.656268209483219455314033425809, −27.43908882134198998977789010362, −26.633404762275850355469690068121, −26.147087894200418497530050641203, −25.01727110726965690494568970920, −24.04613082831790073788368358085, −22.80752648556988450626645722875, −21.663708590567451983819659970, −20.86139450617605822773312952941, −19.17348927184739990493177248833, −18.47117807726241158064400300465, −17.23981911302371827026730225312, −16.11006360000392664548428211305, −15.09984248178458319095789804138, −14.15716083420301047381055985074, −13.68786787335919558092789101965, −11.1175601475517495887522056714, −10.2884644818284913827293249184, −9.00820109514020164050422545178, −8.062371214412944620215614385519, −6.82983359490346506829501536390, −5.34234988896683282635149314884, −4.13470349325384689445114146723, −2.32288849252184122943655505922,
1.37735981651585476007870269764, 2.30078684446266344072202094276, 4.061720330492741533795995078043, 5.491199178644841171755047342312, 7.7835080842956350796158527851, 8.35617553744417313346182043124, 9.52245840180590202976449572619, 10.831988066192814252130311898041, 12.41354723129172430503307212953, 12.82358994300882385398881352312, 13.98126484550951787150035590079, 15.2648378249132869020310237595, 17.3431871527459588120858846246, 17.713542239505452621605499123396, 18.92519429776716998201275325944, 20.08967167483498541416052634559, 20.71383232146060077413679563473, 21.48488109250140618609414259404, 23.21204002511532681774248864790, 24.15340878447569015861395783662, 25.34122882439391574492383092722, 26.09597664186657213402420844809, 27.65810413867372337244980793019, 28.15748775221285271930264430728, 29.48021746988522000313779159628