L(s) = 1 | + (0.114 + 0.993i)2-s + (−0.409 − 0.912i)3-s + (−0.973 + 0.227i)4-s + (−0.0383 + 0.999i)5-s + (0.859 − 0.511i)6-s + (0.606 − 0.795i)7-s + (−0.338 − 0.941i)8-s + (−0.665 + 0.746i)9-s + (−0.997 + 0.0765i)10-s + (−0.927 − 0.373i)11-s + (0.606 + 0.795i)12-s + (0.543 − 0.839i)13-s + (0.859 + 0.511i)14-s + (0.927 − 0.373i)15-s + (0.896 − 0.443i)16-s + (0.988 − 0.152i)17-s + ⋯ |
L(s) = 1 | + (0.114 + 0.993i)2-s + (−0.409 − 0.912i)3-s + (−0.973 + 0.227i)4-s + (−0.0383 + 0.999i)5-s + (0.859 − 0.511i)6-s + (0.606 − 0.795i)7-s + (−0.338 − 0.941i)8-s + (−0.665 + 0.746i)9-s + (−0.997 + 0.0765i)10-s + (−0.927 − 0.373i)11-s + (0.606 + 0.795i)12-s + (0.543 − 0.839i)13-s + (0.859 + 0.511i)14-s + (0.927 − 0.373i)15-s + (0.896 − 0.443i)16-s + (0.988 − 0.152i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.143407372 - 0.2954417697i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.143407372 - 0.2954417697i\) |
\(L(1)\) |
\(\approx\) |
\(0.9172976869 + 0.09053228764i\) |
\(L(1)\) |
\(\approx\) |
\(0.9172976869 + 0.09053228764i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 83 | \( 1 \) |
good | 2 | \( 1 + (0.114 + 0.993i)T \) |
| 3 | \( 1 + (-0.409 - 0.912i)T \) |
| 5 | \( 1 + (-0.0383 + 0.999i)T \) |
| 7 | \( 1 + (0.606 - 0.795i)T \) |
| 11 | \( 1 + (-0.927 - 0.373i)T \) |
| 13 | \( 1 + (0.543 - 0.839i)T \) |
| 17 | \( 1 + (0.988 - 0.152i)T \) |
| 19 | \( 1 + (0.771 - 0.636i)T \) |
| 23 | \( 1 + (0.817 - 0.575i)T \) |
| 29 | \( 1 + (0.190 - 0.981i)T \) |
| 31 | \( 1 + (0.338 - 0.941i)T \) |
| 37 | \( 1 + (-0.665 - 0.746i)T \) |
| 41 | \( 1 + (-0.114 + 0.993i)T \) |
| 43 | \( 1 + (-0.953 + 0.301i)T \) |
| 47 | \( 1 + (-0.477 + 0.878i)T \) |
| 53 | \( 1 + (-0.477 - 0.878i)T \) |
| 59 | \( 1 + (0.720 - 0.693i)T \) |
| 61 | \( 1 + (-0.264 + 0.964i)T \) |
| 67 | \( 1 + (-0.896 + 0.443i)T \) |
| 71 | \( 1 + (-0.606 - 0.795i)T \) |
| 73 | \( 1 + (0.997 - 0.0765i)T \) |
| 79 | \( 1 + (0.973 - 0.227i)T \) |
| 89 | \( 1 + (0.859 - 0.511i)T \) |
| 97 | \( 1 + (0.859 + 0.511i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−31.09667896238918969532082239501, −29.20221980523824060497363496025, −28.51450968451017839664971197316, −27.83087773717826568108786429883, −26.96446939568869809748153592072, −25.548633854854599389864828623504, −23.858552184023908450749947915496, −23.09709129325240313505096853251, −21.61342326265761190492081890633, −21.07123911833471377949663080797, −20.35750126823795825529800344364, −18.77750853886087943916937795610, −17.693389987167019487040201607296, −16.46438820309351988441144014803, −15.27070189211430372199434619196, −13.92624167095836695532437094888, −12.38125249965532629552386088611, −11.710104330464977547300895403528, −10.41318896771949558766047825092, −9.27073699663180789049489439812, −8.36140159301025145235985113139, −5.43375057825816354364679930672, −4.91303288047175718775254055099, −3.40336377756832791894374582116, −1.453266897161340600119121312597,
0.6381380672325382518483991070, 3.11714348417879759412822186927, 5.11499865599070482217836557875, 6.28490541656471630094150897957, 7.496885782449581618920625387249, 8.05099277119169819527115777639, 10.25044855590536998789758343082, 11.40505819208090909544755680665, 13.11716141409922164549466535838, 13.8466020172542963449578927414, 14.9608504461509313467750561346, 16.32123266761129952216640917073, 17.57959044647672300930171721690, 18.21112387103943708056256148516, 19.16511781072245973807025370407, 20.98570327454558286750665598359, 22.56702246933555259727132629881, 23.156588128328325023050864301868, 24.02223740080996897964118495088, 25.10222229383905908370061626812, 26.17780251551850082475636964779, 27.029561363790030462601050177972, 28.32576242605456570109162747945, 29.89334692878530255380143256805, 30.44074534659209440455021062391