L(s) = 1 | + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (0.623 + 0.781i)5-s + (0.222 + 0.974i)7-s + (0.900 − 0.433i)8-s + (0.222 − 0.974i)10-s + (−0.900 − 0.433i)11-s + (−0.900 − 0.433i)13-s + (0.623 − 0.781i)14-s + (−0.900 − 0.433i)16-s + 17-s + (0.222 − 0.974i)19-s + (−0.900 + 0.433i)20-s + (0.222 + 0.974i)22-s + (−0.222 + 0.974i)25-s + (0.222 + 0.974i)26-s + ⋯ |
L(s) = 1 | + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (0.623 + 0.781i)5-s + (0.222 + 0.974i)7-s + (0.900 − 0.433i)8-s + (0.222 − 0.974i)10-s + (−0.900 − 0.433i)11-s + (−0.900 − 0.433i)13-s + (0.623 − 0.781i)14-s + (−0.900 − 0.433i)16-s + 17-s + (0.222 − 0.974i)19-s + (−0.900 + 0.433i)20-s + (0.222 + 0.974i)22-s + (−0.222 + 0.974i)25-s + (0.222 + 0.974i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9225666130 + 0.4999014903i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9225666130 + 0.4999014903i\) |
\(L(1)\) |
\(\approx\) |
\(0.8223096071 + 0.002819163841i\) |
\(L(1)\) |
\(\approx\) |
\(0.8223096071 + 0.002819163841i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (-0.623 - 0.781i)T \) |
| 5 | \( 1 + (0.623 + 0.781i)T \) |
| 7 | \( 1 + (0.222 + 0.974i)T \) |
| 11 | \( 1 + (-0.900 - 0.433i)T \) |
| 13 | \( 1 + (-0.900 - 0.433i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + (0.222 - 0.974i)T \) |
| 31 | \( 1 + (0.623 + 0.781i)T \) |
| 37 | \( 1 + (0.900 - 0.433i)T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + (-0.623 + 0.781i)T \) |
| 47 | \( 1 + (0.900 + 0.433i)T \) |
| 53 | \( 1 + (0.623 + 0.781i)T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + (0.222 + 0.974i)T \) |
| 67 | \( 1 + (0.900 - 0.433i)T \) |
| 71 | \( 1 + (0.900 + 0.433i)T \) |
| 73 | \( 1 + (0.623 - 0.781i)T \) |
| 79 | \( 1 + (0.900 - 0.433i)T \) |
| 83 | \( 1 + (-0.222 + 0.974i)T \) |
| 89 | \( 1 + (0.623 + 0.781i)T \) |
| 97 | \( 1 + (0.222 - 0.974i)T \) |
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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
−19.94052347104994909632697969636, −18.83579447906352427266152517910, −18.34937103194665600678086773286, −17.37984926859603923716706603749, −16.87842255981520188946886870557, −16.55463045915684554435802648514, −15.58341548006054655221315830841, −14.74192064144332174339948713044, −14.02381223791036037656908291910, −13.4793747374848590149395506578, −12.5906801250162332886582429661, −11.69296621316690508029412423080, −10.48834380607483058848281865462, −9.93522175025766104985837641911, −9.5925748080323852126322110919, −8.35149231150261940085986198847, −7.863228084299530847334409604111, −7.15091591480970426847924398818, −6.2336096459264143955758537705, −5.26965592719827519362494117000, −4.86982001820324998265818327159, −3.87443371241553268383025729706, −2.30379023201325341370158793278, −1.49312826916165330901566680029, −0.50471299985450079125601906844,
1.05447220102206796860860253286, 2.30436950636196371143365415582, 2.706607465382661029122061567376, 3.37224631286275373848707580410, 4.86489263783951866514117486087, 5.458170225907264703772878533324, 6.51453322569106239743290781831, 7.5225553072556625110372892691, 8.08095036087007695883192097014, 9.067539531673908868396650151722, 9.71352523879106547327967809832, 10.41174383200304827890011167643, 11.03958377228472475238261937732, 11.88019764717282297586252624065, 12.527701503399407624611924587610, 13.35093718768368485073922426637, 14.07564545877920147122696226146, 15.013134515771756683524834855625, 15.670459683359692273366793066492, 16.67109377335546053801482202843, 17.423890590019521565996665373875, 18.15617023575665432531267569682, 18.48505777451296936712172889873, 19.23158272259005380848660479884, 19.92905152851872403338573911608