L(s) = 1 | + (0.997 − 0.0770i)2-s + (0.988 − 0.153i)4-s + (−0.998 − 0.0550i)5-s + (0.973 − 0.229i)8-s + (−0.999 + 0.0220i)10-s + (−0.975 + 0.218i)11-s + (0.148 + 0.988i)13-s + (0.952 − 0.303i)16-s + (−0.556 + 0.831i)17-s + (−0.999 − 0.0220i)19-s + (−0.995 + 0.0990i)20-s + (−0.956 + 0.293i)22-s + (−0.329 + 0.944i)23-s + (0.993 + 0.110i)25-s + (0.224 + 0.974i)26-s + ⋯ |
L(s) = 1 | + (0.997 − 0.0770i)2-s + (0.988 − 0.153i)4-s + (−0.998 − 0.0550i)5-s + (0.973 − 0.229i)8-s + (−0.999 + 0.0220i)10-s + (−0.975 + 0.218i)11-s + (0.148 + 0.988i)13-s + (0.952 − 0.303i)16-s + (−0.556 + 0.831i)17-s + (−0.999 − 0.0220i)19-s + (−0.995 + 0.0990i)20-s + (−0.956 + 0.293i)22-s + (−0.329 + 0.944i)23-s + (0.993 + 0.110i)25-s + (0.224 + 0.974i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6923681977 - 1.028500353i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6923681977 - 1.028500353i\) |
\(L(1)\) |
\(\approx\) |
\(1.329607828 - 0.1254237768i\) |
\(L(1)\) |
\(\approx\) |
\(1.329607828 - 0.1254237768i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 191 | \( 1 \) |
good | 2 | \( 1 + (0.997 - 0.0770i)T \) |
| 5 | \( 1 + (-0.998 - 0.0550i)T \) |
| 11 | \( 1 + (-0.975 + 0.218i)T \) |
| 13 | \( 1 + (0.148 + 0.988i)T \) |
| 17 | \( 1 + (-0.556 + 0.831i)T \) |
| 19 | \( 1 + (-0.999 - 0.0220i)T \) |
| 23 | \( 1 + (-0.329 + 0.944i)T \) |
| 29 | \( 1 + (-0.746 - 0.665i)T \) |
| 31 | \( 1 + (0.191 - 0.981i)T \) |
| 37 | \( 1 + (-0.191 - 0.981i)T \) |
| 41 | \( 1 + (-0.879 - 0.475i)T \) |
| 43 | \( 1 + (-0.340 - 0.940i)T \) |
| 47 | \( 1 + (0.815 + 0.578i)T \) |
| 53 | \( 1 + (0.660 - 0.750i)T \) |
| 59 | \( 1 + (-0.421 - 0.906i)T \) |
| 61 | \( 1 + (-0.938 - 0.345i)T \) |
| 67 | \( 1 + (0.884 - 0.466i)T \) |
| 71 | \( 1 + (0.724 - 0.689i)T \) |
| 73 | \( 1 + (0.999 + 0.0110i)T \) |
| 79 | \( 1 + (-0.949 - 0.314i)T \) |
| 83 | \( 1 + (0.652 - 0.757i)T \) |
| 89 | \( 1 + (-0.899 - 0.436i)T \) |
| 97 | \( 1 + (0.0165 - 0.999i)T \) |
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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
−18.59070071752327948015891986881, −18.259792916200124840590103981785, −17.052680115871658686769276335516, −16.41472788601602667939665079847, −15.77964761040053103315025465765, −15.23654872193615420657161999495, −14.803294488077497854756190184674, −13.814173440645318070421885541232, −13.21994762330875942665929006120, −12.51393960226210304446801858428, −12.075333034637026804588260784365, −11.04179785737606719720271778966, −10.793805980854157570492033262998, −10.008291572484474103257371944551, −8.503331336452836384674290462720, −8.26848736299921801085270996230, −7.313019815028314712122128098267, −6.797728470389993066236508071630, −5.892152020370265686143234915317, −5.03520519180867495689293710383, −4.57023787048541793435151812473, −3.664724115251642137741116799808, −2.94499406204571001402421011427, −2.397371482508215997764647184860, −1.059396359707485896379849097079,
0.23958755797422489746366366132, 1.852981216682713495797048354060, 2.23966170863636951757990998745, 3.475110309137653847316098781773, 3.97962660697278643927951922827, 4.57254344448585127877210523903, 5.42038015411790916090908475020, 6.22372533663248040036665142760, 6.97395426741679031385110765561, 7.67208785960961523377393284501, 8.27304934614730094668363953945, 9.23117695878132488756217373744, 10.27391495578684708148856504411, 10.956825491233927471867338945750, 11.42698107960961167016578693353, 12.20115492698241039280613884535, 12.78839145708920266418303669922, 13.427934209932686488739684873379, 14.11534119517680101520694483775, 15.10718998351384958484293938513, 15.34919552074738472518528827905, 15.93893880092168766089403351617, 16.80107422830226657132565730149, 17.31141371605783671973368954545, 18.64543684527615291919566084987