L(s) = 1 | + (−0.491 + 0.870i)2-s + (0.928 + 0.372i)3-s + (−0.516 − 0.856i)4-s + (0.0733 + 0.997i)5-s + (−0.780 + 0.625i)6-s + (0.102 + 0.994i)7-s + (0.999 − 0.0293i)8-s + (0.722 + 0.691i)9-s + (−0.904 − 0.426i)10-s + (−0.957 + 0.289i)11-s + (−0.160 − 0.986i)12-s + (−0.891 + 0.452i)13-s + (−0.916 − 0.399i)14-s + (−0.303 + 0.952i)15-s + (−0.465 + 0.884i)16-s + (−0.863 + 0.504i)17-s + ⋯ |
L(s) = 1 | + (−0.491 + 0.870i)2-s + (0.928 + 0.372i)3-s + (−0.516 − 0.856i)4-s + (0.0733 + 0.997i)5-s + (−0.780 + 0.625i)6-s + (0.102 + 0.994i)7-s + (0.999 − 0.0293i)8-s + (0.722 + 0.691i)9-s + (−0.904 − 0.426i)10-s + (−0.957 + 0.289i)11-s + (−0.160 − 0.986i)12-s + (−0.891 + 0.452i)13-s + (−0.916 − 0.399i)14-s + (−0.303 + 0.952i)15-s + (−0.465 + 0.884i)16-s + (−0.863 + 0.504i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0812 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0812 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.8164686636 + 0.8857743481i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.8164686636 + 0.8857743481i\) |
\(L(1)\) |
\(\approx\) |
\(0.5206542684 + 0.7895287427i\) |
\(L(1)\) |
\(\approx\) |
\(0.5206542684 + 0.7895287427i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 643 | \( 1 \) |
good | 2 | \( 1 + (-0.491 + 0.870i)T \) |
| 3 | \( 1 + (0.928 + 0.372i)T \) |
| 5 | \( 1 + (0.0733 + 0.997i)T \) |
| 7 | \( 1 + (0.102 + 0.994i)T \) |
| 11 | \( 1 + (-0.957 + 0.289i)T \) |
| 13 | \( 1 + (-0.891 + 0.452i)T \) |
| 17 | \( 1 + (-0.863 + 0.504i)T \) |
| 19 | \( 1 + (-0.386 + 0.922i)T \) |
| 23 | \( 1 + (0.798 + 0.601i)T \) |
| 29 | \( 1 + (0.957 - 0.289i)T \) |
| 31 | \( 1 + (-0.303 - 0.952i)T \) |
| 37 | \( 1 + (-0.590 - 0.807i)T \) |
| 41 | \( 1 + (0.815 + 0.578i)T \) |
| 43 | \( 1 + (0.978 + 0.204i)T \) |
| 47 | \( 1 + (-0.102 - 0.994i)T \) |
| 53 | \( 1 + (0.998 + 0.0586i)T \) |
| 59 | \( 1 + (0.928 - 0.372i)T \) |
| 61 | \( 1 + (-0.938 + 0.345i)T \) |
| 67 | \( 1 + (-0.761 - 0.647i)T \) |
| 71 | \( 1 + (0.965 + 0.261i)T \) |
| 73 | \( 1 + (-0.938 + 0.345i)T \) |
| 79 | \( 1 + (0.965 - 0.261i)T \) |
| 83 | \( 1 + (-0.303 + 0.952i)T \) |
| 89 | \( 1 + (0.491 + 0.870i)T \) |
| 97 | \( 1 + (-0.948 - 0.317i)T \) |
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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
−21.66447417066370179251385498410, −20.89602634052943861359372942867, −20.39007679859908471292236007542, −19.687096100105697721076566220304, −19.184248530829456061755811587457, −17.859652839417930285117633332465, −17.51735148085312639981097828930, −16.4053168530779145812644201056, −15.55555772233255206101536959143, −14.18859731935850719332358804342, −13.40750201788914310879957104075, −12.91672995676363524577750485728, −12.1748939351636386362236405399, −10.8586078354438203842933790075, −10.164084880855296741207139639866, −9.11872326355961558457987569076, −8.56796938163170639263913871558, −7.6588866385506006907226072326, −6.96050440926774238183211880794, −4.924865332214113639560073301796, −4.37497351740975976429280340362, −3.05606173613448074744712904811, −2.31949379071544889633693638703, −1.0458526034310586836681183194, −0.3179790183140811415937606092,
2.00042753369500741222202392514, 2.555585306489844830110367278932, 4.01269821453453601006820464906, 5.07165332222872387631212794898, 6.0635587668179000972372656429, 7.15336940441470179046229715902, 7.83134478749589318750698988477, 8.71666498708648859752081621654, 9.56802513792332557040462750285, 10.26297477914133338927413035756, 11.12469466710087488088677868937, 12.62937900750062579818700019192, 13.623879184316690442283772727298, 14.49469408351083705050624974400, 15.14066506364414088059959055364, 15.46929527572573575163961656050, 16.512329885891790125176610026793, 17.70954614575718435549153206367, 18.33203649913790465934307623802, 19.22128560862117689987840434112, 19.50467677549545564394376207583, 20.985747632950909733275024069174, 21.66237978433739088585957270031, 22.48142716030811748050217996392, 23.3774773456088920299678183331