L(s) = 1 | + (−0.401 − 0.915i)2-s + (0.789 − 0.614i)3-s + (−0.677 + 0.735i)4-s + (−0.677 + 0.735i)5-s + (−0.879 − 0.475i)6-s + 7-s + (0.945 + 0.324i)8-s + (0.245 − 0.969i)9-s + (0.945 + 0.324i)10-s + (−0.986 − 0.164i)11-s + (−0.0825 + 0.996i)12-s + (0.789 + 0.614i)13-s + (−0.401 − 0.915i)14-s + (−0.0825 + 0.996i)15-s + (−0.0825 − 0.996i)16-s + (0.546 − 0.837i)17-s + ⋯ |
L(s) = 1 | + (−0.401 − 0.915i)2-s + (0.789 − 0.614i)3-s + (−0.677 + 0.735i)4-s + (−0.677 + 0.735i)5-s + (−0.879 − 0.475i)6-s + 7-s + (0.945 + 0.324i)8-s + (0.245 − 0.969i)9-s + (0.945 + 0.324i)10-s + (−0.986 − 0.164i)11-s + (−0.0825 + 0.996i)12-s + (0.789 + 0.614i)13-s + (−0.401 − 0.915i)14-s + (−0.0825 + 0.996i)15-s + (−0.0825 − 0.996i)16-s + (0.546 − 0.837i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.209 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.209 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8929113659 - 0.7219110450i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8929113659 - 0.7219110450i\) |
\(L(1)\) |
\(\approx\) |
\(0.9281299419 - 0.5030652151i\) |
\(L(1)\) |
\(\approx\) |
\(0.9281299419 - 0.5030652151i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 191 | \( 1 \) |
good | 2 | \( 1 + (-0.401 - 0.915i)T \) |
| 3 | \( 1 + (0.789 - 0.614i)T \) |
| 5 | \( 1 + (-0.677 + 0.735i)T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + (-0.986 - 0.164i)T \) |
| 13 | \( 1 + (0.789 + 0.614i)T \) |
| 17 | \( 1 + (0.546 - 0.837i)T \) |
| 19 | \( 1 + (0.945 - 0.324i)T \) |
| 23 | \( 1 + (0.945 - 0.324i)T \) |
| 29 | \( 1 + (-0.0825 + 0.996i)T \) |
| 31 | \( 1 + (0.245 - 0.969i)T \) |
| 37 | \( 1 + (0.245 + 0.969i)T \) |
| 41 | \( 1 + (-0.401 + 0.915i)T \) |
| 43 | \( 1 + (-0.879 - 0.475i)T \) |
| 47 | \( 1 + (-0.986 - 0.164i)T \) |
| 53 | \( 1 + (-0.986 - 0.164i)T \) |
| 59 | \( 1 + (0.245 - 0.969i)T \) |
| 61 | \( 1 + (0.546 + 0.837i)T \) |
| 67 | \( 1 + (0.546 + 0.837i)T \) |
| 71 | \( 1 + (-0.401 + 0.915i)T \) |
| 73 | \( 1 + (-0.986 + 0.164i)T \) |
| 79 | \( 1 + (-0.0825 - 0.996i)T \) |
| 83 | \( 1 + (0.945 + 0.324i)T \) |
| 89 | \( 1 + (-0.879 + 0.475i)T \) |
| 97 | \( 1 + (0.245 + 0.969i)T \) |
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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
−27.09782699233296191515403149587, −26.47622509672966035773854473928, −25.32955747727084633128014882718, −24.66173042801026603092594624977, −23.673569904483624808494832285257, −22.9086786973985479610818707795, −21.26975051052199802155797898412, −20.61791586325968963069378993572, −19.59221054495241111497835585621, −18.589356601075143330224169693397, −17.4980463886497547486072880, −16.36702535964209647980801401290, −15.577285588680929218539970650673, −14.96483292232290651900352753393, −13.843024329091482009590512390639, −12.85882327841492874354900437460, −11.10112567228661190804219834309, −10.112821571463435775735206918187, −8.86385712629729371176908174002, −8.067231197373386225643134951908, −7.61331390663205956295549439537, −5.4834544603396510935286147429, −4.78511365996479253243600796187, −3.54713509123782100212660093900, −1.434504214961270737835504291921,
1.22243271440988846605226920769, 2.629518212496179441062450901, 3.4368376914094445536646646034, 4.85939108897874235769600168042, 7.02230747917000094313079727684, 7.89126349500940598908358496454, 8.61324314076143678105083156708, 9.927000576848564780051476574334, 11.2412020811132350132839451841, 11.72452525914383052442164854769, 13.120920943560122058598536059329, 13.95897344494597215075659326191, 14.9019690696497561545455615984, 16.21848017938332528784092683198, 17.84202720643579266280413465458, 18.55605443862689855256439117512, 18.92001467255373022867978580038, 20.34223487093969228569400669096, 20.71792518089597231363971591479, 21.84780058209901777173366796619, 23.20842713255312266319360895233, 23.84270639070581518807126049071, 25.21079883030972692655246843456, 26.29872227883871705019491608185, 26.76326514803028552694595748696