L(s) = 1 | + (0.623 + 0.781i)3-s + (−0.900 + 0.433i)5-s + (−0.623 − 0.781i)7-s + (−0.222 + 0.974i)9-s + (−0.222 − 0.974i)11-s + (−0.222 − 0.974i)13-s + (−0.900 − 0.433i)15-s − 17-s + (0.623 − 0.781i)19-s + (0.222 − 0.974i)21-s + (0.900 + 0.433i)23-s + (0.623 − 0.781i)25-s + (−0.900 + 0.433i)27-s + (−0.900 + 0.433i)31-s + (0.623 − 0.781i)33-s + ⋯ |
L(s) = 1 | + (0.623 + 0.781i)3-s + (−0.900 + 0.433i)5-s + (−0.623 − 0.781i)7-s + (−0.222 + 0.974i)9-s + (−0.222 − 0.974i)11-s + (−0.222 − 0.974i)13-s + (−0.900 − 0.433i)15-s − 17-s + (0.623 − 0.781i)19-s + (0.222 − 0.974i)21-s + (0.900 + 0.433i)23-s + (0.623 − 0.781i)25-s + (−0.900 + 0.433i)27-s + (−0.900 + 0.433i)31-s + (0.623 − 0.781i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3472940677 - 0.4789682607i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3472940677 - 0.4789682607i\) |
\(L(1)\) |
\(\approx\) |
\(0.8241761964 + 0.03736267892i\) |
\(L(1)\) |
\(\approx\) |
\(0.8241761964 + 0.03736267892i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 \) |
good | 3 | \( 1 + (0.623 + 0.781i)T \) |
| 5 | \( 1 + (-0.900 + 0.433i)T \) |
| 7 | \( 1 + (-0.623 - 0.781i)T \) |
| 11 | \( 1 + (-0.222 - 0.974i)T \) |
| 13 | \( 1 + (-0.222 - 0.974i)T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + (0.623 - 0.781i)T \) |
| 23 | \( 1 + (0.900 + 0.433i)T \) |
| 31 | \( 1 + (-0.900 + 0.433i)T \) |
| 37 | \( 1 + (0.222 - 0.974i)T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + (-0.900 - 0.433i)T \) |
| 47 | \( 1 + (-0.222 - 0.974i)T \) |
| 53 | \( 1 + (-0.900 + 0.433i)T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + (-0.623 - 0.781i)T \) |
| 67 | \( 1 + (0.222 - 0.974i)T \) |
| 71 | \( 1 + (0.222 + 0.974i)T \) |
| 73 | \( 1 + (0.900 + 0.433i)T \) |
| 79 | \( 1 + (-0.222 + 0.974i)T \) |
| 83 | \( 1 + (-0.623 + 0.781i)T \) |
| 89 | \( 1 + (0.900 - 0.433i)T \) |
| 97 | \( 1 + (-0.623 + 0.781i)T \) |
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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
−29.02673236964777028303300920649, −28.61458787814843831697522660906, −27.17370462916231997370011156486, −26.141094189501026243305813737133, −25.150608378938258014463321694931, −24.29960557480285602374874039217, −23.35918890842910229247297793484, −22.322047284933026905923039744528, −20.75830748419909248039908482264, −19.93549192970869269386077890260, −19.00384975752735099627996382698, −18.253591619717378990531908418667, −16.737641539373155897955781870123, −15.51200920085857680956160963985, −14.72175473453386521539831666306, −13.211964814792954210197330274100, −12.40063992638583393345544147822, −11.569003438240850132829372483842, −9.548035246684427855798826222183, −8.69081020556594289204518808160, −7.49318046573369210773882759757, −6.490981537620674502000702687213, −4.67770370902070374115940603541, −3.20024183239200755387506856500, −1.78946448208466202230015421962,
0.21994717479969991399289956928, 2.97192005447728523564761946414, 3.665885424482630632251590190647, 5.08407275344337047787550092376, 6.94605157407818084975795281286, 8.0257575518852407006940375393, 9.20663641176627155820318897662, 10.56579193947465528521948935637, 11.18465062301931133429171241561, 13.02078099225303841178190032653, 13.96836878699544527958419464035, 15.278589629020330116506356265753, 15.86554579168447122077083112002, 16.96894864859131450142402215915, 18.56987273717288695109990974911, 19.88664769246738571531492331721, 19.98463009604897336699562544478, 21.62892709911352385163217080039, 22.46721097134394051105960930613, 23.45771895285679084865631867106, 24.7115783334603054271701684918, 25.97203388104659192448783803669, 26.822237856701110853022986812710, 27.202667933704949431970164217479, 28.60672676505921829028145979524