| L(s) = 1 | + (0.104 − 0.994i)2-s + (1.46 − 0.921i)3-s + (−0.978 − 0.207i)4-s − 2.54·5-s + (−0.763 − 1.55i)6-s + (−0.402 − 1.23i)7-s + (−0.309 + 0.951i)8-s + (1.30 − 2.70i)9-s + (−0.265 + 2.52i)10-s + (−5.67 − 1.20i)11-s + (−1.62 + 0.596i)12-s + (−2.07 + 1.50i)13-s + (−1.27 + 0.270i)14-s + (−3.72 + 2.34i)15-s + (0.913 + 0.406i)16-s + (0.163 + 0.181i)17-s + ⋯ |
| L(s) = 1 | + (0.0739 − 0.703i)2-s + (0.846 − 0.532i)3-s + (−0.489 − 0.103i)4-s − 1.13·5-s + (−0.311 − 0.634i)6-s + (−0.152 − 0.468i)7-s + (−0.109 + 0.336i)8-s + (0.433 − 0.901i)9-s + (−0.0840 + 0.799i)10-s + (−1.70 − 0.363i)11-s + (−0.469 + 0.172i)12-s + (−0.575 + 0.418i)13-s + (−0.340 + 0.0723i)14-s + (−0.962 + 0.604i)15-s + (0.228 + 0.101i)16-s + (0.0395 + 0.0439i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.110286 + 0.835405i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.110286 + 0.835405i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.104 + 0.994i)T \) |
| 3 | \( 1 + (-1.46 + 0.921i)T \) |
| 31 | \( 1 + (5.45 + 1.09i)T \) |
| good | 5 | \( 1 + 2.54T + 5T^{2} \) |
| 7 | \( 1 + (0.402 + 1.23i)T + (-5.66 + 4.11i)T^{2} \) |
| 11 | \( 1 + (5.67 + 1.20i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (2.07 - 1.50i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.163 - 0.181i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-3.44 + 1.53i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (0.448 + 0.498i)T + (-2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (0.0534 - 0.509i)T + (-28.3 - 6.02i)T^{2} \) |
| 37 | \( 1 + (-4.89 + 8.48i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.54 + 1.84i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-2.54 - 1.84i)T + (13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (8.64 + 3.84i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-4.36 + 0.928i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-3.09 - 1.37i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (-5.20 - 9.00i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + 0.605T + 67T^{2} \) |
| 71 | \( 1 + (-4.47 + 0.951i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (-6.72 + 7.47i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (1.20 - 3.70i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (0.699 - 0.311i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (3.04 + 9.38i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-11.8 + 13.1i)T + (-10.1 - 96.4i)T^{2} \) |
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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
−10.34396131592264213789286576691, −9.443179288347439464901353605992, −8.429717016347219559443431128990, −7.67167265944446457629699009904, −7.13139969890923969311315190388, −5.43671802975044887748940992832, −4.18844179060251106508551712597, −3.31340838605003828926494320340, −2.30817354914992172472335118700, −0.40799750427758241991918681652,
2.63430522466381383099683246269, 3.60204642916697205160110458240, 4.78133383602923876296350674090, 5.45603538293459009479662314049, 7.11527451762483662019351499800, 7.973248241856116575882493677792, 8.132677126489662971230124763517, 9.447905952481861280680850931301, 10.10464521022748708589582333864, 11.13117619193206048540413815196
