| L(s) = 1 | + (1.18 − 1.32i)2-s + (0.564 − 0.251i)3-s + (−0.121 − 1.15i)4-s + (2.71 − 0.577i)5-s + (0.339 − 1.04i)6-s + (1.20 + 0.878i)8-s + (−1.75 + 1.94i)9-s + (2.46 − 4.27i)10-s + (2.12 + 2.55i)11-s + (−0.358 − 0.620i)12-s + (−1.32 − 4.08i)13-s + (1.38 − 1.00i)15-s + (4.86 − 1.03i)16-s + (−1.84 − 2.04i)17-s + (0.486 + 4.62i)18-s + (−0.202 + 1.92i)19-s + ⋯ |
| L(s) = 1 | + (0.841 − 0.934i)2-s + (0.325 − 0.145i)3-s + (−0.0605 − 0.576i)4-s + (1.21 − 0.258i)5-s + (0.138 − 0.426i)6-s + (0.427 + 0.310i)8-s + (−0.583 + 0.648i)9-s + (0.780 − 1.35i)10-s + (0.639 + 0.768i)11-s + (−0.103 − 0.179i)12-s + (−0.367 − 1.13i)13-s + (0.358 − 0.260i)15-s + (1.21 − 0.258i)16-s + (−0.447 − 0.496i)17-s + (0.114 + 1.09i)18-s + (−0.0465 + 0.442i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.48690 - 1.54253i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.48690 - 1.54253i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 + (-2.12 - 2.55i)T \) |
| good | 2 | \( 1 + (-1.18 + 1.32i)T + (-0.209 - 1.98i)T^{2} \) |
| 3 | \( 1 + (-0.564 + 0.251i)T + (2.00 - 2.22i)T^{2} \) |
| 5 | \( 1 + (-2.71 + 0.577i)T + (4.56 - 2.03i)T^{2} \) |
| 13 | \( 1 + (1.32 + 4.08i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (1.84 + 2.04i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (0.202 - 1.92i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (2.18 + 3.78i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (6.98 - 5.07i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.196 - 0.0417i)T + (28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (-0.945 - 0.421i)T + (24.7 + 27.4i)T^{2} \) |
| 41 | \( 1 + (7.77 + 5.64i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 4.70T + 43T^{2} \) |
| 47 | \( 1 + (-1.36 + 12.9i)T + (-45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (-3.81 - 0.810i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (-0.894 - 8.51i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (-0.967 + 0.205i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (-2.70 + 4.69i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.623 - 1.91i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-1.04 - 9.91i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (-4.21 + 4.67i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (0.531 - 1.63i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-7.65 - 13.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.58 - 11.0i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.61533847996198284389105324871, −10.10269561163226116966925853622, −9.058989046796945599294489310335, −8.087569817046825505355070879223, −6.96962647325674332853007384599, −5.53684128590668138764060290601, −5.07573696559240797303004247906, −3.72049698497363558462303213376, −2.48995239832190968315923825476, −1.80877551448858275107634423002,
1.86660643378036025716901648140, 3.42240771189310940214061105384, 4.45265678997103986034258947133, 5.74406230180543577935039909192, 6.20685930102906312742719143143, 6.93386111129172827700043973547, 8.228852890273927825934769906690, 9.338798905377375081844034938441, 9.751263019535830928101469956495, 11.06537625102721209293947060728
