L(s) = 1 | + (0.957 + 0.288i)2-s + (0.744 − 0.667i)3-s + (0.833 + 0.551i)4-s + (−0.934 − 0.357i)5-s + (0.905 − 0.424i)6-s + (−0.997 − 0.0729i)7-s + (0.639 + 0.768i)8-s + (0.109 − 0.994i)9-s + (−0.791 − 0.611i)10-s + (−0.976 + 0.217i)11-s + (0.989 − 0.145i)12-s + (0.905 + 0.424i)13-s + (−0.934 − 0.357i)14-s + (−0.934 + 0.357i)15-s + (0.391 + 0.920i)16-s + (−0.457 + 0.889i)17-s + ⋯ |
L(s) = 1 | + (0.957 + 0.288i)2-s + (0.744 − 0.667i)3-s + (0.833 + 0.551i)4-s + (−0.934 − 0.357i)5-s + (0.905 − 0.424i)6-s + (−0.997 − 0.0729i)7-s + (0.639 + 0.768i)8-s + (0.109 − 0.994i)9-s + (−0.791 − 0.611i)10-s + (−0.976 + 0.217i)11-s + (0.989 − 0.145i)12-s + (0.905 + 0.424i)13-s + (−0.934 − 0.357i)14-s + (−0.934 + 0.357i)15-s + (0.391 + 0.920i)16-s + (−0.457 + 0.889i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.058457697 + 0.09876883809i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.058457697 + 0.09876883809i\) |
\(L(1)\) |
\(\approx\) |
\(1.937886423 + 0.008631098613i\) |
\(L(1)\) |
\(\approx\) |
\(1.937886423 + 0.008631098613i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 + (0.957 + 0.288i)T \) |
| 3 | \( 1 + (0.744 - 0.667i)T \) |
| 5 | \( 1 + (-0.934 - 0.357i)T \) |
| 7 | \( 1 + (-0.997 - 0.0729i)T \) |
| 11 | \( 1 + (-0.976 + 0.217i)T \) |
| 13 | \( 1 + (0.905 + 0.424i)T \) |
| 17 | \( 1 + (-0.457 + 0.889i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (0.989 + 0.145i)T \) |
| 29 | \( 1 + (0.109 + 0.994i)T \) |
| 31 | \( 1 + (0.905 - 0.424i)T \) |
| 37 | \( 1 + (0.520 - 0.853i)T \) |
| 41 | \( 1 + (0.109 - 0.994i)T \) |
| 47 | \( 1 + (0.109 + 0.994i)T \) |
| 53 | \( 1 + (0.639 - 0.768i)T \) |
| 59 | \( 1 + (-0.181 - 0.983i)T \) |
| 61 | \( 1 + (-0.997 - 0.0729i)T \) |
| 67 | \( 1 + (0.957 + 0.288i)T \) |
| 71 | \( 1 + (0.109 - 0.994i)T \) |
| 73 | \( 1 + (0.833 - 0.551i)T \) |
| 79 | \( 1 + (0.252 + 0.967i)T \) |
| 83 | \( 1 + (-0.997 - 0.0729i)T \) |
| 89 | \( 1 + (-0.0365 + 0.999i)T \) |
| 97 | \( 1 + (0.905 - 0.424i)T \) |
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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
−19.99255207632197575909924241771, −19.882626231243670211029282295093, −18.66813953607661254389840705947, −18.56426313716836797350174997919, −16.6776578413891593440348494802, −16.00436541386268762596547694206, −15.485789212261895331001056355303, −15.27332295325139262395874883552, −14.09418549949332219964478130421, −13.42841170747741692356282064789, −13.01993112030990858057238368693, −11.86663216340077473278279461576, −11.222100615700177776079055502826, −10.43214963033446018120487716335, −9.86670100863354183617660147224, −8.82570432660283305613915434589, −7.88381358114246468059049618208, −7.17739741882025864904377450958, −6.25466218641259144340861505538, −5.23620439154494967158424287907, −4.49064157996623564086691233073, −3.624216341875290479869758833859, −2.86737265940562794511246975590, −2.72338693025416225114494094992, −0.86925538940399406785175267436,
0.987795270815332863670946990851, 2.21200023795082841396922613900, 3.18851535106331583292856208778, 3.612895336892476682945396319505, 4.506117575628786782505462736030, 5.58206152895921130203005132721, 6.50125618497943388244927023620, 7.14701082041238110451205691231, 7.82988034411301238807899832415, 8.5561754329402310973141921124, 9.35929824096186487976081272370, 10.66634744556676398488160438492, 11.398668465390804286213149702412, 12.44614715348497232559683890152, 12.736522328541435683459724443847, 13.408646693387778413091808371213, 14.05797474385562958876381643218, 15.097708412863430904932317571987, 15.62495961158899844896474831176, 16.08386130955649459851760847998, 16.98334523880466329510273369258, 18.06243018022039152397059033727, 18.93161489443144792329710362594, 19.53994326244419829914267232788, 20.17636890890177605274433578133