L(s) = 1 | + (−0.809 + 0.587i)3-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)11-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)17-s + (0.809 + 0.587i)19-s + (0.309 + 0.951i)23-s + (0.309 + 0.951i)27-s + (0.809 − 0.587i)29-s + (−0.809 − 0.587i)31-s + (−0.809 − 0.587i)33-s + (0.309 − 0.951i)37-s + (−0.309 − 0.951i)39-s + (−0.309 + 0.951i)41-s − 43-s + ⋯ |
L(s) = 1 | + (−0.809 + 0.587i)3-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)11-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)17-s + (0.809 + 0.587i)19-s + (0.309 + 0.951i)23-s + (0.309 + 0.951i)27-s + (0.809 − 0.587i)29-s + (−0.809 − 0.587i)31-s + (−0.809 − 0.587i)33-s + (0.309 − 0.951i)37-s + (−0.309 − 0.951i)39-s + (−0.309 + 0.951i)41-s − 43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2977571462 + 0.7520487048i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2977571462 + 0.7520487048i\) |
\(L(1)\) |
\(\approx\) |
\(0.7221326428 + 0.2848084530i\) |
\(L(1)\) |
\(\approx\) |
\(0.7221326428 + 0.2848084530i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 + (0.309 + 0.951i)T \) |
| 13 | \( 1 + (-0.309 + 0.951i)T \) |
| 17 | \( 1 + (-0.809 - 0.587i)T \) |
| 19 | \( 1 + (0.809 + 0.587i)T \) |
| 23 | \( 1 + (0.309 + 0.951i)T \) |
| 29 | \( 1 + (0.809 - 0.587i)T \) |
| 31 | \( 1 + (-0.809 - 0.587i)T \) |
| 37 | \( 1 + (0.309 - 0.951i)T \) |
| 41 | \( 1 + (-0.309 + 0.951i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (0.809 - 0.587i)T \) |
| 53 | \( 1 + (-0.809 + 0.587i)T \) |
| 59 | \( 1 + (-0.309 + 0.951i)T \) |
| 61 | \( 1 + (0.309 + 0.951i)T \) |
| 67 | \( 1 + (0.809 + 0.587i)T \) |
| 71 | \( 1 + (0.809 - 0.587i)T \) |
| 73 | \( 1 + (0.309 + 0.951i)T \) |
| 79 | \( 1 + (0.809 - 0.587i)T \) |
| 83 | \( 1 + (-0.809 - 0.587i)T \) |
| 89 | \( 1 + (-0.309 - 0.951i)T \) |
| 97 | \( 1 + (-0.809 + 0.587i)T \) |
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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
−20.29390462296056126111371854971, −19.74796010755318946298843395153, −18.869488665124927950295610367151, −18.22225312663555221942211487673, −17.47867990837437368902450336718, −16.88881028330962561046353982577, −16.04721854418320410979514396578, −15.36712249263788001374951083125, −14.24285257347146488830091186045, −13.548958981450446141005987003238, −12.723688680467698090858529428407, −12.186515825917688231649869199740, −11.06007503949513122688428066667, −10.83936824358043157660242348495, −9.75782820326849368874081970137, −8.60660299288891289535537854827, −8.01070520881479578264696279517, −6.88164620538490012037529109449, −6.417575898804318473661725013007, −5.37589407224220674217814905370, −4.82446781784669452384276413783, −3.51053705957424648359127597708, −2.559169628854838907872072921683, −1.37062261957208526915186251596, −0.38815363708615925878766840913,
1.21647354476384935953513521472, 2.30706932129685775159708270826, 3.628245026863965205592461305974, 4.4038423295302707176007987372, 5.07776171030792914860048967882, 6.018244214105354419777466910562, 6.90468748771934280398847527177, 7.515242880953383312528039051, 8.94777995252210802366149736025, 9.568105025608355694451460684216, 10.13174168699869502841236833466, 11.28426731703924517484672053368, 11.7004972843655607360898004518, 12.44064095659550703110419111856, 13.438520473646062802752278978941, 14.35844951454028312242777591275, 15.15441926059234374406327576180, 15.8247803797557388230982540666, 16.596510230706871359265947213774, 17.239379854190582181960699905692, 17.987215754896765635759899877426, 18.599223939332900244785417676968, 19.77764236468783732848635347846, 20.32653444961343834387076714365, 21.25944534893185014560591759387