L(s) = 1 | + 0.537·2-s + (0.5 + 0.866i)3-s − 1.71·4-s + (−0.249 − 0.431i)5-s + (0.268 + 0.465i)6-s + 1.57·7-s − 1.99·8-s + (−0.499 + 0.866i)9-s + (−0.134 − 0.232i)10-s + (2.88 + 4.99i)11-s + (−0.855 − 1.48i)12-s + (−2.90 + 5.02i)13-s + 0.848·14-s + (0.249 − 0.431i)15-s + 2.34·16-s + (−0.250 − 0.433i)17-s + ⋯ |
L(s) = 1 | + 0.380·2-s + (0.288 + 0.499i)3-s − 0.855·4-s + (−0.111 − 0.193i)5-s + (0.109 + 0.190i)6-s + 0.596·7-s − 0.705·8-s + (−0.166 + 0.288i)9-s + (−0.0423 − 0.0734i)10-s + (0.869 + 1.50i)11-s + (−0.246 − 0.427i)12-s + (−0.805 + 1.39i)13-s + 0.226·14-s + (0.0643 − 0.111i)15-s + 0.587·16-s + (−0.0607 − 0.105i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0980 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0980 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.06398 + 0.964331i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06398 + 0.964331i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 157 | \( 1 + (-7.31 - 10.1i)T \) |
good | 2 | \( 1 - 0.537T + 2T^{2} \) |
| 5 | \( 1 + (0.249 + 0.431i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 - 1.57T + 7T^{2} \) |
| 11 | \( 1 + (-2.88 - 4.99i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.90 - 5.02i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.250 + 0.433i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.13 - 1.96i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 2.83T + 23T^{2} \) |
| 29 | \( 1 + 2.88T + 29T^{2} \) |
| 31 | \( 1 + (4.96 - 8.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.25 + 5.63i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 0.670T + 41T^{2} \) |
| 43 | \( 1 + (2.94 - 5.10i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.922 - 1.59i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.29 + 9.17i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 3.44T + 59T^{2} \) |
| 61 | \( 1 + (2.92 + 5.07i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 - 9.17T + 67T^{2} \) |
| 71 | \( 1 + (-6.95 + 12.0i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (3.00 + 5.20i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 7.23T + 79T^{2} \) |
| 83 | \( 1 + (-8.01 + 13.8i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.43 - 2.49i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (9.65 - 16.7i)T + (-48.5 - 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.38027417518345308212657683402, −10.07741053133020018433559266383, −9.354529167888538548933011240024, −8.837998601512139482066016833579, −7.63162101294201076302617902918, −6.62168896902060766240168612324, −4.98666608069124587356646068366, −4.62814624776117835147953795673, −3.66699288568082426845988270188, −1.90010144945009161100985876012,
0.826437001509581221924586120753, 2.90020601268781641482625178427, 3.81747657352646502379420095915, 5.18517341618635048521203414643, 5.89962576688555030483255683228, 7.24670816106993851007085646307, 8.182843075073558422845592239079, 8.864566454178777661861034431726, 9.765392355597804588628895813461, 11.05243111548270153637425969983