L(s) = 1 | + (−0.900 − 0.433i)2-s + (−0.222 − 0.974i)3-s + (0.623 + 0.781i)4-s + (−0.222 − 0.974i)5-s + (−0.222 + 0.974i)6-s + (−0.222 − 0.974i)8-s + (−0.900 + 0.433i)9-s + (−0.222 + 0.974i)10-s + (−0.900 − 0.433i)11-s + (0.623 − 0.781i)12-s + (−0.900 − 0.433i)13-s + (−0.900 + 0.433i)15-s + (−0.222 + 0.974i)16-s + (0.623 − 0.781i)17-s + 18-s + 19-s + ⋯ |
L(s) = 1 | + (−0.900 − 0.433i)2-s + (−0.222 − 0.974i)3-s + (0.623 + 0.781i)4-s + (−0.222 − 0.974i)5-s + (−0.222 + 0.974i)6-s + (−0.222 − 0.974i)8-s + (−0.900 + 0.433i)9-s + (−0.222 + 0.974i)10-s + (−0.900 − 0.433i)11-s + (0.623 − 0.781i)12-s + (−0.900 − 0.433i)13-s + (−0.900 + 0.433i)15-s + (−0.222 + 0.974i)16-s + (0.623 − 0.781i)17-s + 18-s + 19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1570800948 - 0.4268373620i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1570800948 - 0.4268373620i\) |
\(L(1)\) |
\(\approx\) |
\(0.4366845057 - 0.3825441013i\) |
\(L(1)\) |
\(\approx\) |
\(0.4366845057 - 0.3825441013i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
good | 2 | \( 1 + (-0.900 - 0.433i)T \) |
| 3 | \( 1 + (-0.222 - 0.974i)T \) |
| 5 | \( 1 + (-0.222 - 0.974i)T \) |
| 11 | \( 1 + (-0.900 - 0.433i)T \) |
| 13 | \( 1 + (-0.900 - 0.433i)T \) |
| 17 | \( 1 + (0.623 - 0.781i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (0.623 + 0.781i)T \) |
| 29 | \( 1 + (0.623 - 0.781i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (0.623 - 0.781i)T \) |
| 41 | \( 1 + (-0.222 - 0.974i)T \) |
| 43 | \( 1 + (-0.222 + 0.974i)T \) |
| 47 | \( 1 + (-0.900 - 0.433i)T \) |
| 53 | \( 1 + (0.623 + 0.781i)T \) |
| 59 | \( 1 + (-0.222 + 0.974i)T \) |
| 61 | \( 1 + (0.623 - 0.781i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (0.623 + 0.781i)T \) |
| 73 | \( 1 + (-0.900 + 0.433i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (-0.900 + 0.433i)T \) |
| 89 | \( 1 + (-0.900 + 0.433i)T \) |
| 97 | \( 1 + T \) |
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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
−34.20071135563324623740035470969, −33.5317886314063611596578998506, −32.34096247146441889670034148592, −30.89847796934645522520167015232, −29.23564306114711642774304216754, −28.346722925327816024040611171110, −27.02768989486612229440781973286, −26.484949728525398833522118732750, −25.49667130536254073774493272494, −23.77540328673930719136339140191, −22.68721944438723484399487505576, −21.32266906771563983246996079616, −19.98870248429834940166339868895, −18.71368516128555029382072606383, −17.55185138112648092126002908951, −16.33830144923573849041085251373, −15.215173683105178221869488634361, −14.4320396594566081730135319926, −11.7631775228370578818945767035, −10.48726256328192917339560235123, −9.78619198348865543619779416373, −8.06985933497139805237288880742, −6.64776176162004323808696295551, −5.044400320231324786207707825397, −2.83973610442772228754210029031,
0.871228361333598126613073440567, 2.76062525074196651097524011736, 5.374016248721491520259061091213, 7.387924779242156129839925566160, 8.21682653824405354047776723879, 9.72873650042050863621263757421, 11.46914214076433103532338447528, 12.38588567579592584913567933935, 13.49066419367160554525274440230, 15.83567054342101594863191466743, 16.966999506901093452462007399803, 17.98143934269544131113897285764, 19.1621208495271326287567150667, 20.09854402443800123057272093322, 21.26674951413903569932902683326, 23.04926476654701159201524885834, 24.46278817166579249104830290813, 25.073941949109800427360976231025, 26.62783909877850526921322068051, 27.838628417971457212340872253529, 28.92972136445549302897187940370, 29.50987986808113683545497197625, 30.91614338290827164799929050346, 31.90668780734970529604202540631, 33.89774573435192908608024634253