| L(s) = 1 | + (−0.971 − 0.662i)2-s + (1.44 + 1.34i)3-s + (−0.225 − 0.575i)4-s + (−0.154 + 0.0475i)5-s + (−0.515 − 2.25i)6-s + (2.49 + 0.876i)7-s + (−0.684 + 3.00i)8-s + (0.0659 + 0.880i)9-s + (0.181 + 0.0559i)10-s + (−0.385 + 5.14i)11-s + (0.444 − 1.13i)12-s + (0.900 − 0.433i)13-s + (−1.84 − 2.50i)14-s + (−0.286 − 0.138i)15-s + (1.74 − 1.62i)16-s + (−4.86 + 0.733i)17-s + ⋯ |
| L(s) = 1 | + (−0.686 − 0.468i)2-s + (0.833 + 0.773i)3-s + (−0.112 − 0.287i)4-s + (−0.0689 + 0.0212i)5-s + (−0.210 − 0.922i)6-s + (0.943 + 0.331i)7-s + (−0.242 + 1.06i)8-s + (0.0219 + 0.293i)9-s + (0.0573 + 0.0176i)10-s + (−0.116 + 1.55i)11-s + (0.128 − 0.327i)12-s + (0.249 − 0.120i)13-s + (−0.493 − 0.669i)14-s + (−0.0740 − 0.0356i)15-s + (0.436 − 0.405i)16-s + (−1.17 + 0.177i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.21942 + 0.484815i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21942 + 0.484815i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.49 - 0.876i)T \) |
| 13 | \( 1 + (-0.900 + 0.433i)T \) |
| good | 2 | \( 1 + (0.971 + 0.662i)T + (0.730 + 1.86i)T^{2} \) |
| 3 | \( 1 + (-1.44 - 1.34i)T + (0.224 + 2.99i)T^{2} \) |
| 5 | \( 1 + (0.154 - 0.0475i)T + (4.13 - 2.81i)T^{2} \) |
| 11 | \( 1 + (0.385 - 5.14i)T + (-10.8 - 1.63i)T^{2} \) |
| 17 | \( 1 + (4.86 - 0.733i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (-2.26 - 3.92i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.27 - 0.192i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (-2.10 - 2.64i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (-4.23 + 7.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.63 + 6.71i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (1.41 - 6.21i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (-2.77 - 12.1i)T + (-38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (2.44 + 1.66i)T + (17.1 + 43.7i)T^{2} \) |
| 53 | \( 1 + (-0.410 - 1.04i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-0.454 - 0.140i)T + (48.7 + 33.2i)T^{2} \) |
| 61 | \( 1 + (1.73 - 4.41i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (0.0180 - 0.0312i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.16 + 6.48i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (1.48 - 1.01i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (5.95 + 10.3i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-13.4 - 6.46i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (0.583 + 7.78i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 - 5.41T + 97T^{2} \) |
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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
−10.45303783196751990678936680556, −9.626406696962729176960826411401, −9.262511409411653798267344305613, −8.281766460626222163808323015026, −7.66090402408794415077527638122, −6.09848664460028104712044521828, −4.86755376083103416735961634267, −4.18871061621359308692546167799, −2.62256909913231825523225906504, −1.65151268779331419780648272990,
0.879604635678906885791495298747, 2.50716328302394733894963964627, 3.66070171193306449639680506385, 4.93775132259173769719707776679, 6.44770949129360417198407923377, 7.19604235117467347814201807426, 8.068434453339555714891150733337, 8.532649894567673545390406673303, 9.035073849361616767482199087423, 10.41704443549041878007339928754
