L(s) = 1 | + i·3-s + i·7-s − 9-s + 11-s + i·13-s + i·17-s + 19-s − 21-s + i·23-s − i·27-s − 29-s + 31-s + i·33-s − 39-s + 41-s + ⋯ |
L(s) = 1 | + i·3-s + i·7-s − 9-s + 11-s + i·13-s + i·17-s + 19-s − 21-s + i·23-s − i·27-s − 29-s + 31-s + i·33-s − 39-s + 41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4161273951 + 1.464829610i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4161273951 + 1.464829610i\) |
\(L(1)\) |
\(\approx\) |
\(0.9031452314 + 0.6412356449i\) |
\(L(1)\) |
\(\approx\) |
\(0.9031452314 + 0.6412356449i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 3 | \( 1 + T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| 17 | \( 1 \) |
| 19 | \( 1 + iT \) |
| 23 | \( 1 \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 \) |
| 41 | \( 1 \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 \) |
| 53 | \( 1 \) |
| 59 | \( 1 \) |
| 61 | \( 1 + iT \) |
| 67 | \( 1 \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 \) |
| 89 | \( 1 + iT \) |
| 97 | \( 1 \) |
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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
−20.32591123182763270841016501368, −19.6473266470308427147394076373, −18.87406518245815372685538039150, −17.99776191662624484327956226007, −17.51061691462599265980480116819, −16.7145702376677028954230802290, −16.035112051907190658723969538982, −14.7465064955624874926274212202, −14.22973070801913416824123152045, −13.45644221085132164971819444667, −12.884301554474505245333541071589, −11.89614871479613752595773313635, −11.37859430419103030237364501832, −10.39942240919467652877705081856, −9.5226615979570096523259202653, −8.59096545611020577260402540031, −7.66289119856724833105724416385, −7.1725234005608543825851073362, −6.36229328751180785633903847193, −5.480699863912272524984003202384, −4.42985207460449764473382160949, −3.371492035733630982685071123461, −2.569505594509637274348755125759, −1.265664950819015169510394845458, −0.641540535522301554039404838474,
1.44793663606853065704975310059, 2.47580949501013067625561606782, 3.57623411938214563695067137242, 4.14314028136176539599861704874, 5.2275956935351423372934727188, 5.87442987018099598797454137905, 6.72567704700964140456543718132, 7.95772159645824757959975644563, 8.87384142982389289317130306783, 9.359383608110354963040411170054, 9.9982831440603782649271073240, 11.23780197604042629012600001932, 11.60475828581893328536427706066, 12.37630901021512045685507999458, 13.557687817315782421341557874422, 14.412947764066485693937403163994, 14.91036377925254939250153521672, 15.759139788875553924650058996417, 16.30191287214280469138216216526, 17.21602124687175638598928575632, 17.73778169040339271495618760550, 19.01038227623975324475514910674, 19.35640683987228156379279406239, 20.33401835886887004785519125881, 21.131852503979515065510136184229