L(s) = 1 | + 2-s + (0.893 − 0.448i)3-s + 4-s + (−0.993 + 0.116i)5-s + (0.893 − 0.448i)6-s + (0.597 + 0.802i)7-s + 8-s + (0.597 − 0.802i)9-s + (−0.993 + 0.116i)10-s + (−0.993 − 0.116i)11-s + (0.893 − 0.448i)12-s + (−0.993 + 0.116i)13-s + (0.597 + 0.802i)14-s + (−0.835 + 0.549i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯ |
L(s) = 1 | + 2-s + (0.893 − 0.448i)3-s + 4-s + (−0.993 + 0.116i)5-s + (0.893 − 0.448i)6-s + (0.597 + 0.802i)7-s + 8-s + (0.597 − 0.802i)9-s + (−0.993 + 0.116i)10-s + (−0.993 − 0.116i)11-s + (0.893 − 0.448i)12-s + (−0.993 + 0.116i)13-s + (0.597 + 0.802i)14-s + (−0.835 + 0.549i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(4.367854679 - 0.3248358642i\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.367854679 - 0.3248358642i\) |
\(L(1)\) |
\(\approx\) |
\(2.361528596 - 0.1595135400i\) |
\(L(1)\) |
\(\approx\) |
\(2.361528596 - 0.1595135400i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + T \) |
| 3 | \( 1 + (0.893 - 0.448i)T \) |
| 5 | \( 1 + (-0.993 + 0.116i)T \) |
| 7 | \( 1 + (0.597 + 0.802i)T \) |
| 11 | \( 1 + (-0.993 - 0.116i)T \) |
| 13 | \( 1 + (-0.993 + 0.116i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.173 + 0.984i)T \) |
| 23 | \( 1 + (0.766 - 0.642i)T \) |
| 29 | \( 1 + (0.597 + 0.802i)T \) |
| 31 | \( 1 + (0.973 - 0.230i)T \) |
| 41 | \( 1 + (0.766 + 0.642i)T \) |
| 43 | \( 1 + (0.173 - 0.984i)T \) |
| 47 | \( 1 + (0.893 + 0.448i)T \) |
| 53 | \( 1 + (-0.0581 + 0.998i)T \) |
| 59 | \( 1 + (-0.835 + 0.549i)T \) |
| 61 | \( 1 + (-0.286 - 0.957i)T \) |
| 67 | \( 1 + (0.893 + 0.448i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (0.597 - 0.802i)T \) |
| 79 | \( 1 + (-0.835 + 0.549i)T \) |
| 83 | \( 1 + (-0.686 + 0.727i)T \) |
| 89 | \( 1 + (0.396 - 0.918i)T \) |
| 97 | \( 1 + (0.973 + 0.230i)T \) |
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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
−18.8853540838029845219461627036, −17.54402979952358910984239257003, −17.024399533532738985856528747888, −16.020898804976989904209057375108, −15.53733456087197172973560751921, −15.11304561326548039489750978109, −14.476712330771743643276743608954, −13.721599920780734087728103418618, −13.141088175826727253890548784861, −12.545771073667698185459815393983, −11.61013695583426681002530872588, −10.91147405053862291492607047215, −10.46633075338624370856429907160, −9.60272584194390205968917846596, −8.439807578646089233142096551038, −7.89304249914141530627644737770, −7.36757787086694497258026604460, −6.76507147940932926805511363727, −5.347759062304727125536378043326, −4.66663800221996831527708486664, −4.38019581602609793884697026543, −3.51593948404445557581397844440, −2.77441121069401828482094175304, −2.1364283838335350221867812723, −0.90714327895085680364014318828,
0.93968977716959423072512332364, 2.12335860230806910195891765034, 2.74723831645721406291005589568, 3.125154756656535766768649798003, 4.32933806135436511527295887821, 4.74108251810680931279830026718, 5.5941380102229564339573641824, 6.632892757073092919683558587062, 7.3121605700948062656706312896, 7.86417586063396971981311323357, 8.41235489527351219561969082938, 9.32317669960366044761478078784, 10.36494060589254274336967556992, 11.098173901807747781818136356973, 11.90846040702871207659137780936, 12.42296336677678958950287800263, 12.79127104658245542642514529193, 13.98274180059539539065933560118, 14.21152236850023051260630775114, 15.08898084452963670197868726924, 15.44477112427069618694770546434, 15.98955169212348929572702648444, 16.92146065537988871795811413318, 18.01379909330461495927087700005, 18.80189411790669562869785041989