L(s) = 1 | + (0.935 − 0.354i)2-s + (0.992 + 0.120i)3-s + (0.748 − 0.663i)4-s + (−0.239 + 0.970i)5-s + (0.970 − 0.239i)6-s + (0.354 + 0.935i)7-s + (0.464 − 0.885i)8-s + (0.970 + 0.239i)9-s + (0.120 + 0.992i)10-s + (−0.568 − 0.822i)11-s + (0.822 − 0.568i)12-s + (−0.748 − 0.663i)13-s + (0.663 + 0.748i)14-s + (−0.354 + 0.935i)15-s + (0.120 − 0.992i)16-s + (−0.885 + 0.464i)17-s + ⋯ |
L(s) = 1 | + (0.935 − 0.354i)2-s + (0.992 + 0.120i)3-s + (0.748 − 0.663i)4-s + (−0.239 + 0.970i)5-s + (0.970 − 0.239i)6-s + (0.354 + 0.935i)7-s + (0.464 − 0.885i)8-s + (0.970 + 0.239i)9-s + (0.120 + 0.992i)10-s + (−0.568 − 0.822i)11-s + (0.822 − 0.568i)12-s + (−0.748 − 0.663i)13-s + (0.663 + 0.748i)14-s + (−0.354 + 0.935i)15-s + (0.120 − 0.992i)16-s + (−0.885 + 0.464i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.398935701 - 0.1166999320i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.398935701 - 0.1166999320i\) |
\(L(1)\) |
\(\approx\) |
\(2.303288861 - 0.1132617516i\) |
\(L(1)\) |
\(\approx\) |
\(2.303288861 - 0.1132617516i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 53 | \( 1 \) |
good | 2 | \( 1 + (0.935 - 0.354i)T \) |
| 3 | \( 1 + (0.992 + 0.120i)T \) |
| 5 | \( 1 + (-0.239 + 0.970i)T \) |
| 7 | \( 1 + (0.354 + 0.935i)T \) |
| 11 | \( 1 + (-0.568 - 0.822i)T \) |
| 13 | \( 1 + (-0.748 - 0.663i)T \) |
| 17 | \( 1 + (-0.885 + 0.464i)T \) |
| 19 | \( 1 + (0.663 - 0.748i)T \) |
| 23 | \( 1 - iT \) |
| 29 | \( 1 + (-0.568 + 0.822i)T \) |
| 31 | \( 1 + (0.822 + 0.568i)T \) |
| 37 | \( 1 + (-0.120 + 0.992i)T \) |
| 41 | \( 1 + (-0.822 + 0.568i)T \) |
| 43 | \( 1 + (-0.120 - 0.992i)T \) |
| 47 | \( 1 + (-0.970 + 0.239i)T \) |
| 59 | \( 1 + (0.970 - 0.239i)T \) |
| 61 | \( 1 + (0.464 - 0.885i)T \) |
| 67 | \( 1 + (0.663 + 0.748i)T \) |
| 71 | \( 1 + (0.992 - 0.120i)T \) |
| 73 | \( 1 + (0.464 + 0.885i)T \) |
| 79 | \( 1 + (-0.935 - 0.354i)T \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 + (0.885 - 0.464i)T \) |
| 97 | \( 1 + (-0.970 - 0.239i)T \) |
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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
−33.01161232914738152718153371571, −31.65496936143035485907024556326, −31.30005209483440627706538510303, −30.03960144342662431570410604027, −28.884966796669080444170240869618, −27.02299717584110313749938145761, −26.063509021722378088059591464120, −24.72224320533929349332715637774, −24.12377131480185969848164919269, −22.9686624166951325958625250889, −21.20344001358278173366365335494, −20.48082569292972535585649480173, −19.65346258766410801419687670046, −17.52206970661628208552305545348, −16.21459360837240778388280118499, −15.108764253189463930706090014507, −13.85201626577344961497781314128, −13.04422539724473121877352767845, −11.75209816871288466438297988511, −9.689559067388522909880805716, −7.988752158853460092607275414214, −7.18586491694581651241347212714, −4.89378173269170792366775769296, −3.92877855102882561553832800766, −1.97289795196752793674060497126,
2.38715394433710642836174020563, 3.22452092238083069635378367436, 5.020940802055195509470665923191, 6.77266646310682236437119662921, 8.31733182302766345410589325731, 10.128178026851173458497170809035, 11.31774483967039799784121607668, 12.82374604979849407707636419045, 14.06445534109558183265536982177, 15.058349595473160509330468354842, 15.71209764123208505486206494514, 18.323948602591788745284863762049, 19.26191358864989120639541270766, 20.36334450761878011958893044602, 21.728302751881387527867399340512, 22.218529184770917826287635666520, 24.01062777541410644414117743570, 24.821100177624043304785062436632, 26.13590490445421324986311315222, 27.26176788893948490577033530456, 28.821806367141590312092691334043, 30.1435865522640938478737143398, 30.896810297462069029238302900769, 31.7103069658313842453840313780, 32.705969443808085090768835622951