| L(s) = 1 | + (8.01 + 2.91i)2-s + (10.1 − 11.8i)3-s + (31.1 + 26.1i)4-s + (−3.30 + 18.7i)5-s + (116. − 64.8i)6-s + (−6.74 + 5.65i)7-s + (37.1 + 64.2i)8-s + (−35.6 − 240. i)9-s + (−81.1 + 140. i)10-s + (81.7 + 463. i)11-s + (626. − 101. i)12-s + (−883. + 321. i)13-s + (−70.5 + 25.6i)14-s + (187. + 229. i)15-s + (−116. − 659. i)16-s + (380. − 659. i)17-s + ⋯ |
| L(s) = 1 | + (1.41 + 0.515i)2-s + (0.653 − 0.757i)3-s + (0.974 + 0.817i)4-s + (−0.0591 + 0.335i)5-s + (1.31 − 0.735i)6-s + (−0.0520 + 0.0436i)7-s + (0.204 + 0.354i)8-s + (−0.146 − 0.989i)9-s + (−0.256 + 0.444i)10-s + (0.203 + 1.15i)11-s + (1.25 − 0.203i)12-s + (−1.45 + 0.527i)13-s + (−0.0961 + 0.0350i)14-s + (0.215 + 0.263i)15-s + (−0.113 − 0.644i)16-s + (0.319 − 0.553i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.190i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.26544 + 0.314149i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.26544 + 0.314149i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-10.1 + 11.8i)T \) |
| good | 2 | \( 1 + (-8.01 - 2.91i)T + (24.5 + 20.5i)T^{2} \) |
| 5 | \( 1 + (3.30 - 18.7i)T + (-2.93e3 - 1.06e3i)T^{2} \) |
| 7 | \( 1 + (6.74 - 5.65i)T + (2.91e3 - 1.65e4i)T^{2} \) |
| 11 | \( 1 + (-81.7 - 463. i)T + (-1.51e5 + 5.50e4i)T^{2} \) |
| 13 | \( 1 + (883. - 321. i)T + (2.84e5 - 2.38e5i)T^{2} \) |
| 17 | \( 1 + (-380. + 659. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-40.7 - 70.5i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (2.60e3 + 2.18e3i)T + (1.11e6 + 6.33e6i)T^{2} \) |
| 29 | \( 1 + (-558. - 203. i)T + (1.57e7 + 1.31e7i)T^{2} \) |
| 31 | \( 1 + (-7.07e3 - 5.93e3i)T + (4.97e6 + 2.81e7i)T^{2} \) |
| 37 | \( 1 + (-392. + 679. i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + (-1.77e4 + 6.47e3i)T + (8.87e7 - 7.44e7i)T^{2} \) |
| 43 | \( 1 + (-1.36e3 - 7.76e3i)T + (-1.38e8 + 5.02e7i)T^{2} \) |
| 47 | \( 1 + (-2.90e3 + 2.43e3i)T + (3.98e7 - 2.25e8i)T^{2} \) |
| 53 | \( 1 + 2.84e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-358. + 2.03e3i)T + (-6.71e8 - 2.44e8i)T^{2} \) |
| 61 | \( 1 + (-1.77e4 + 1.49e4i)T + (1.46e8 - 8.31e8i)T^{2} \) |
| 67 | \( 1 + (-1.51e3 + 551. i)T + (1.03e9 - 8.67e8i)T^{2} \) |
| 71 | \( 1 + (-3.47e4 + 6.02e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (-1.54e4 - 2.67e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (9.41e4 + 3.42e4i)T + (2.35e9 + 1.97e9i)T^{2} \) |
| 83 | \( 1 + (8.76e4 + 3.18e4i)T + (3.01e9 + 2.53e9i)T^{2} \) |
| 89 | \( 1 + (1.75e4 + 3.04e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + (-2.17e4 - 1.23e5i)T + (-8.06e9 + 2.93e9i)T^{2} \) |
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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
−15.81124744146591450815317341135, −14.48802015494252065206308819976, −14.27078288511419949728581924628, −12.63915993883754252514617190580, −12.09274756091837813169771886124, −9.611869228998927817530363105259, −7.49688843182185127339872476849, −6.57049979250353215918852577383, −4.57977087273315718014796751095, −2.66420024953358269769053918982,
2.79354250084406004528726882082, 4.22686285777494430660980835665, 5.61072595544171631706578498235, 8.215688904640030092152076637620, 9.936384438113297207218743117814, 11.36117758852813466081687534024, 12.67844141971945889662429670925, 13.85951653277550000392783376574, 14.68144554873194771549755458317, 15.79411701420922347282683972759
