L(s) = 1 | + (−0.884 − 0.466i)2-s + (0.951 + 0.309i)3-s + (0.564 + 0.825i)4-s + (−0.696 − 0.717i)6-s + (−0.113 − 0.993i)8-s + (0.809 + 0.587i)9-s + (0.281 + 0.959i)12-s + (−0.336 + 0.941i)13-s + (−0.362 + 0.931i)16-s + (0.226 − 0.974i)17-s + (−0.441 − 0.897i)18-s + (0.516 + 0.856i)19-s + (−0.540 − 0.841i)23-s + (0.198 − 0.980i)24-s + (0.736 − 0.676i)26-s + (0.587 + 0.809i)27-s + ⋯ |
L(s) = 1 | + (−0.884 − 0.466i)2-s + (0.951 + 0.309i)3-s + (0.564 + 0.825i)4-s + (−0.696 − 0.717i)6-s + (−0.113 − 0.993i)8-s + (0.809 + 0.587i)9-s + (0.281 + 0.959i)12-s + (−0.336 + 0.941i)13-s + (−0.362 + 0.931i)16-s + (0.226 − 0.974i)17-s + (−0.441 − 0.897i)18-s + (0.516 + 0.856i)19-s + (−0.540 − 0.841i)23-s + (0.198 − 0.980i)24-s + (0.736 − 0.676i)26-s + (0.587 + 0.809i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.780958586 + 0.1348428754i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.780958586 + 0.1348428754i\) |
\(L(1)\) |
\(\approx\) |
\(1.072004575 + 0.01203637037i\) |
\(L(1)\) |
\(\approx\) |
\(1.072004575 + 0.01203637037i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.884 - 0.466i)T \) |
| 3 | \( 1 + (0.951 + 0.309i)T \) |
| 13 | \( 1 + (-0.336 + 0.941i)T \) |
| 17 | \( 1 + (0.226 - 0.974i)T \) |
| 19 | \( 1 + (0.516 + 0.856i)T \) |
| 23 | \( 1 + (-0.540 - 0.841i)T \) |
| 29 | \( 1 + (0.921 + 0.389i)T \) |
| 31 | \( 1 + (-0.610 - 0.791i)T \) |
| 37 | \( 1 + (0.999 - 0.0285i)T \) |
| 41 | \( 1 + (0.985 + 0.170i)T \) |
| 43 | \( 1 + (0.909 - 0.415i)T \) |
| 47 | \( 1 + (0.441 - 0.897i)T \) |
| 53 | \( 1 + (-0.931 + 0.362i)T \) |
| 59 | \( 1 + (-0.985 + 0.170i)T \) |
| 61 | \( 1 + (0.466 + 0.884i)T \) |
| 67 | \( 1 + (0.989 - 0.142i)T \) |
| 71 | \( 1 + (0.0855 - 0.996i)T \) |
| 73 | \( 1 + (-0.633 + 0.774i)T \) |
| 79 | \( 1 + (0.870 - 0.491i)T \) |
| 83 | \( 1 + (-0.0570 - 0.998i)T \) |
| 89 | \( 1 + (-0.654 - 0.755i)T \) |
| 97 | \( 1 + (-0.980 - 0.198i)T \) |
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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
−18.20511611127299600636361938610, −17.75273828903709070439616306947, −17.27272707854748433635301707297, −16.17777564711466390364290802173, −15.59993535472831350581815982858, −15.1653566587442402069018268024, −14.29752656009261889082284778754, −13.94195636281198806063357279216, −12.858369826664452067377520588449, −12.40320787269147181131337156508, −11.29951598482704916367159888926, −10.64312804548002861016848545194, −9.76479088188418158173635787239, −9.42244818704869635081325267065, −8.55586292177465954591109420050, −7.87079912019688900308513592468, −7.568072498355008041557753625641, −6.649321489414515788709703544255, −5.95856162884246829130391713436, −5.10796541520566133238965738827, −4.08876295624071951857719814251, −3.04820915962649485484508633636, −2.47329362767994022703520406907, −1.503817779548705775527816753802, −0.774919481882442271564788128802,
0.82903471652671280621897303202, 1.83868253433018962562026646388, 2.47954450941910303546755564978, 3.15387615271283840804329660893, 4.062735870840069069510947757346, 4.59168454970732112895194239822, 5.87611459102250751470451868791, 6.870296958337689845405151300262, 7.52421618901924678535999404214, 8.04633404228854897628923621601, 8.91277500879482377932544304786, 9.39878984907907484134395479048, 9.95925161366592786946599035027, 10.64984079552404298903088255868, 11.47031508944910379692733568840, 12.19825529863511628609900818962, 12.784548143702702575862144212620, 13.80065154308129183328249332633, 14.24803257288372693397528529797, 15.02462659315086388674520886300, 16.00902552115832926059269836772, 16.289232968517892186361725068422, 16.93391390815769130549963923151, 18.00580110362167058478042268965, 18.5276061206886971906881238082