L(s) = 1 | + (−0.615 − 0.788i)2-s + (−0.719 + 0.694i)3-s + (−0.241 + 0.970i)4-s + (0.990 + 0.139i)6-s + (−0.913 − 0.406i)7-s + (0.913 − 0.406i)8-s + (0.0348 − 0.999i)9-s + (−0.5 − 0.866i)12-s + (−0.882 + 0.469i)13-s + (0.241 + 0.970i)14-s + (−0.882 − 0.469i)16-s + (−0.0348 − 0.999i)17-s + (−0.809 + 0.587i)18-s + (0.939 − 0.342i)21-s + (−0.766 + 0.642i)23-s + (−0.374 + 0.927i)24-s + ⋯ |
L(s) = 1 | + (−0.615 − 0.788i)2-s + (−0.719 + 0.694i)3-s + (−0.241 + 0.970i)4-s + (0.990 + 0.139i)6-s + (−0.913 − 0.406i)7-s + (0.913 − 0.406i)8-s + (0.0348 − 0.999i)9-s + (−0.5 − 0.866i)12-s + (−0.882 + 0.469i)13-s + (0.241 + 0.970i)14-s + (−0.882 − 0.469i)16-s + (−0.0348 − 0.999i)17-s + (−0.809 + 0.587i)18-s + (0.939 − 0.342i)21-s + (−0.766 + 0.642i)23-s + (−0.374 + 0.927i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2305152884 + 0.01472589237i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2305152884 + 0.01472589237i\) |
\(L(1)\) |
\(\approx\) |
\(0.4028282049 - 0.06499293371i\) |
\(L(1)\) |
\(\approx\) |
\(0.4028282049 - 0.06499293371i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.615 - 0.788i)T \) |
| 3 | \( 1 + (-0.719 + 0.694i)T \) |
| 7 | \( 1 + (-0.913 - 0.406i)T \) |
| 13 | \( 1 + (-0.882 + 0.469i)T \) |
| 17 | \( 1 + (-0.0348 - 0.999i)T \) |
| 23 | \( 1 + (-0.766 + 0.642i)T \) |
| 29 | \( 1 + (-0.961 + 0.275i)T \) |
| 31 | \( 1 + (-0.669 + 0.743i)T \) |
| 37 | \( 1 + (-0.809 + 0.587i)T \) |
| 41 | \( 1 + (0.719 - 0.694i)T \) |
| 43 | \( 1 + (-0.766 - 0.642i)T \) |
| 47 | \( 1 + (-0.559 - 0.829i)T \) |
| 53 | \( 1 + (0.848 + 0.529i)T \) |
| 59 | \( 1 + (-0.559 + 0.829i)T \) |
| 61 | \( 1 + (-0.374 - 0.927i)T \) |
| 67 | \( 1 + (-0.939 - 0.342i)T \) |
| 71 | \( 1 + (-0.848 + 0.529i)T \) |
| 73 | \( 1 + (-0.438 - 0.898i)T \) |
| 79 | \( 1 + (-0.990 + 0.139i)T \) |
| 83 | \( 1 + (0.978 - 0.207i)T \) |
| 89 | \( 1 + (-0.173 - 0.984i)T \) |
| 97 | \( 1 + (-0.615 - 0.788i)T \) |
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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
−21.714311233985437697528287978807, −20.183537087266490629749847756, −19.42809041557813803580871881433, −18.96808830048360121581656996531, −18.09799907185596273076821549868, −17.509041527104888507361532845491, −16.62843622748510869991238881750, −16.23136837773927122193824309878, −15.16481281444428303914100692469, −14.51231462672505173433691716687, −13.293880309142088891377062848340, −12.78176483509331998575107596139, −11.86813759291984471015492724836, −10.796676236828396916358660843557, −10.08796561899959078265587966473, −9.27642558259133592027553395230, −8.18885395478857947130354237533, −7.52259441884832364390856747179, −6.62911850884809195577655756649, −5.96456046630957226414509444716, −5.34742893056703488905640847210, −4.16459494763612183152370253361, −2.54853416266484736764131885075, −1.57927539749527187935342765207, −0.22525362040206930007465444206,
0.27888912749628293984209211249, 1.65907878722395121125078602548, 2.97896871482449420362515020320, 3.72181745877159964682464541879, 4.601587874105765686601577413855, 5.5858969685042213219884994245, 6.88930612122256627795832383554, 7.384387215283125036359092063055, 8.89415000752196106249945268177, 9.47499872571537068302036497788, 10.130994867596525049581134969662, 10.788811653877930782699413368897, 11.82112521157934136544456686223, 12.189691663613958112173498022742, 13.22725501762088138807226371005, 14.08512472666842274760270312964, 15.3276091708242272998950699673, 16.253252956753306034827644049474, 16.636082640382106670494439285391, 17.425450738653082956906438835722, 18.2079605222302561890641106620, 19.01497567404311072895869258739, 19.93547899238734563425907486713, 20.40203754051404230837234789064, 21.4127394202946089777607455799