L(s) = 1 | + (−0.974 − 0.222i)2-s + (−0.433 − 0.900i)3-s + (0.900 + 0.433i)4-s + (0.222 − 0.974i)5-s + (0.222 + 0.974i)6-s + (−0.900 + 0.433i)7-s + (−0.781 − 0.623i)8-s + (−0.623 + 0.781i)9-s + (−0.433 + 0.900i)10-s + (−0.781 + 0.623i)11-s − i·12-s + (−0.623 − 0.781i)13-s + (0.974 − 0.222i)14-s + (−0.974 + 0.222i)15-s + (0.623 + 0.781i)16-s + i·17-s + ⋯ |
L(s) = 1 | + (−0.974 − 0.222i)2-s + (−0.433 − 0.900i)3-s + (0.900 + 0.433i)4-s + (0.222 − 0.974i)5-s + (0.222 + 0.974i)6-s + (−0.900 + 0.433i)7-s + (−0.781 − 0.623i)8-s + (−0.623 + 0.781i)9-s + (−0.433 + 0.900i)10-s + (−0.781 + 0.623i)11-s − i·12-s + (−0.623 − 0.781i)13-s + (0.974 − 0.222i)14-s + (−0.974 + 0.222i)15-s + (0.623 + 0.781i)16-s + i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.03893022907 - 0.2945361561i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.03893022907 - 0.2945361561i\) |
\(L(1)\) |
\(\approx\) |
\(0.3747250616 - 0.2511309087i\) |
\(L(1)\) |
\(\approx\) |
\(0.3747250616 - 0.2511309087i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 \) |
good | 2 | \( 1 + (-0.974 - 0.222i)T \) |
| 3 | \( 1 + (-0.433 - 0.900i)T \) |
| 5 | \( 1 + (0.222 - 0.974i)T \) |
| 7 | \( 1 + (-0.900 + 0.433i)T \) |
| 11 | \( 1 + (-0.781 + 0.623i)T \) |
| 13 | \( 1 + (-0.623 - 0.781i)T \) |
| 17 | \( 1 + iT \) |
| 19 | \( 1 + (0.433 - 0.900i)T \) |
| 23 | \( 1 + (-0.222 - 0.974i)T \) |
| 31 | \( 1 + (-0.974 - 0.222i)T \) |
| 37 | \( 1 + (-0.781 - 0.623i)T \) |
| 41 | \( 1 - iT \) |
| 43 | \( 1 + (0.974 - 0.222i)T \) |
| 47 | \( 1 + (0.781 - 0.623i)T \) |
| 53 | \( 1 + (-0.222 + 0.974i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + (-0.433 - 0.900i)T \) |
| 67 | \( 1 + (-0.623 + 0.781i)T \) |
| 71 | \( 1 + (-0.623 - 0.781i)T \) |
| 73 | \( 1 + (-0.974 + 0.222i)T \) |
| 79 | \( 1 + (0.781 + 0.623i)T \) |
| 83 | \( 1 + (-0.900 - 0.433i)T \) |
| 89 | \( 1 + (-0.974 - 0.222i)T \) |
| 97 | \( 1 + (-0.433 + 0.900i)T \) |
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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
−37.80993381257266007001975821627, −36.46960502124592209044403414400, −35.008586385972894814041117190877, −33.943576809860669899419047960466, −33.262254115807369595485829812974, −31.71972518357945938458926289390, −29.38735919294798782692708859755, −29.05268673290717849849720695740, −27.226945801886639338215726681246, −26.52134535464006100913263987891, −25.59789200548448145166254240641, −23.605109310255890352744603238731, −22.278173584955518879505979603704, −20.898215721725029759565808195954, −19.32358314422320525374585276070, −18.08164008228776669357575914508, −16.62969732897592530304161592763, −15.747003021916387875952157815752, −14.22083851511268109240505224622, −11.548331018847100495150821730577, −10.32166071270391729715886245053, −9.45973420770702444627571995304, −7.236135164049806406728716369542, −5.81118686132454743263652222646, −3.11052327318947713514380790737,
0.2862934905105097622415294941, 2.30697289594641798048655483506, 5.69474229033383494236138712733, 7.37199325865796530868825297488, 8.817102143627068953062667097247, 10.384437039107422299995041945529, 12.335457616553437763722478223536, 12.881093689766006580809061459344, 15.6912475455025026781435499883, 16.99562651617503798887169364400, 17.99049740616847144593320782810, 19.34121122079636210907478682492, 20.35801547798586845634103174551, 22.13517852382507452923648244378, 23.95026749090851111165224916931, 25.01209183772276175636572737841, 26.01858014764433045097600224549, 28.05554372004965076718262632396, 28.62418795508065674265655458023, 29.61490271416393009660347178927, 31.0516154965372715801311961486, 32.73570361008079025482551567695, 34.45580871507419549027526719744, 35.2287473100929020355826505757, 36.32884897925973751989944978013