| L(s) = 1 | + (−0.626 + 0.429i)2-s + (−0.0920 − 1.32i)3-s + (−0.514 + 1.32i)4-s + (−0.0317 − 0.194i)5-s + (0.626 + 0.791i)6-s + (−3.04 − 0.789i)7-s + (−0.595 − 2.53i)8-s + (1.22 − 0.170i)9-s + (0.103 + 0.108i)10-s + (6.33 − 0.146i)11-s + (1.80 + 0.559i)12-s + (2.59 − 4.18i)13-s + (2.24 − 0.810i)14-s + (−0.255 + 0.0600i)15-s + (−0.645 − 0.588i)16-s + (−2.80 − 3.02i)17-s + ⋯ |
| L(s) = 1 | + (−0.442 + 0.303i)2-s + (−0.0531 − 0.765i)3-s + (−0.257 + 0.663i)4-s + (−0.0142 − 0.0871i)5-s + (0.255 + 0.323i)6-s + (−1.14 − 0.298i)7-s + (−0.210 − 0.894i)8-s + (0.406 − 0.0567i)9-s + (0.0327 + 0.0342i)10-s + (1.90 − 0.0441i)11-s + (0.521 + 0.161i)12-s + (0.718 − 1.16i)13-s + (0.599 − 0.216i)14-s + (−0.0659 + 0.0155i)15-s + (−0.161 − 0.147i)16-s + (−0.680 − 0.732i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.683 + 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.683 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.823857 - 0.356986i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.823857 - 0.356986i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (2.80 + 3.02i)T \) |
| good | 2 | \( 1 + (0.626 - 0.429i)T + (0.722 - 1.86i)T^{2} \) |
| 3 | \( 1 + (0.0920 + 1.32i)T + (-2.97 + 0.414i)T^{2} \) |
| 5 | \( 1 + (0.0317 + 0.194i)T + (-4.74 + 1.58i)T^{2} \) |
| 7 | \( 1 + (3.04 + 0.789i)T + (6.11 + 3.40i)T^{2} \) |
| 11 | \( 1 + (-6.33 + 0.146i)T + (10.9 - 0.508i)T^{2} \) |
| 13 | \( 1 + (-2.59 + 4.18i)T + (-5.79 - 11.6i)T^{2} \) |
| 19 | \( 1 + (0.0287 - 0.0419i)T + (-6.86 - 17.7i)T^{2} \) |
| 23 | \( 1 + (-0.161 + 0.623i)T + (-20.0 - 11.1i)T^{2} \) |
| 29 | \( 1 + (3.80 - 3.97i)T + (-1.33 - 28.9i)T^{2} \) |
| 31 | \( 1 + (-0.697 - 0.969i)T + (-9.85 + 29.3i)T^{2} \) |
| 37 | \( 1 + (-0.985 + 1.87i)T + (-20.9 - 30.5i)T^{2} \) |
| 41 | \( 1 + (-5.63 + 6.47i)T + (-5.66 - 40.6i)T^{2} \) |
| 43 | \( 1 + (-1.28 - 0.0594i)T + (42.8 + 3.96i)T^{2} \) |
| 47 | \( 1 + (6.91 - 5.22i)T + (12.8 - 45.2i)T^{2} \) |
| 53 | \( 1 + (-1.13 - 8.10i)T + (-50.9 + 14.5i)T^{2} \) |
| 59 | \( 1 + (10.8 - 6.03i)T + (31.0 - 50.1i)T^{2} \) |
| 61 | \( 1 + (-1.06 + 0.841i)T + (13.9 - 59.3i)T^{2} \) |
| 67 | \( 1 + (5.40 - 1.00i)T + (62.4 - 24.2i)T^{2} \) |
| 71 | \( 1 + (3.53 - 2.07i)T + (34.5 - 62.0i)T^{2} \) |
| 73 | \( 1 + (-8.73 + 4.09i)T + (46.6 - 56.1i)T^{2} \) |
| 79 | \( 1 + (-9.13 + 5.95i)T + (31.9 - 72.2i)T^{2} \) |
| 83 | \( 1 + (-1.93 - 5.77i)T + (-66.2 + 50.0i)T^{2} \) |
| 89 | \( 1 + (1.12 + 1.82i)T + (-39.6 + 79.6i)T^{2} \) |
| 97 | \( 1 + (5.20 + 8.86i)T + (-47.2 + 84.7i)T^{2} \) |
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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
−12.00138399778478218948264941798, −10.68220519437776258262461423223, −9.394840688630256363432966534366, −8.946267195213577012421113985953, −7.66333276446097916290833282304, −6.82903371090101937364812676257, −6.26781159129717428141677315814, −4.20849071788599957968981484053, −3.20883368291132551445921679790, −0.894885977340526765511154197820,
1.62714822486278376518717959142, 3.66831441362585211366712785593, 4.54676806706406625858134700420, 6.16858377482970370390163999855, 6.68420389535225879938641084900, 8.687089977232994864303274687103, 9.424845062822378221957749860713, 9.709368103615661238198173445482, 10.94817642086411932620835377414, 11.53636006031035565281193468625
