| L(s) = 1 | + (1.02 + 0.860i)2-s + (0.817 − 1.52i)3-s + (−0.0359 − 0.204i)4-s + (2.15 − 0.862i)6-s + (0.0601 − 0.340i)7-s + (1.47 − 2.55i)8-s + (−1.66 − 2.49i)9-s + (−0.377 + 0.137i)11-s + (−0.341 − 0.111i)12-s + (−0.575 + 0.483i)13-s + (0.355 − 0.297i)14-s + (3.32 − 1.21i)16-s + (0.670 + 1.16i)17-s + (0.442 − 3.99i)18-s + (1.87 − 3.24i)19-s + ⋯ |
| L(s) = 1 | + (0.725 + 0.608i)2-s + (0.471 − 0.881i)3-s + (−0.0179 − 0.102i)4-s + (0.878 − 0.352i)6-s + (0.0227 − 0.128i)7-s + (0.522 − 0.904i)8-s + (−0.554 − 0.832i)9-s + (−0.113 + 0.0414i)11-s + (−0.0984 − 0.0323i)12-s + (−0.159 + 0.133i)13-s + (0.0948 − 0.0796i)14-s + (0.832 − 0.302i)16-s + (0.162 + 0.281i)17-s + (0.104 − 0.940i)18-s + (0.429 − 0.744i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 + 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.532 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.13173 - 1.17740i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.13173 - 1.17740i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.817 + 1.52i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (-1.02 - 0.860i)T + (0.347 + 1.96i)T^{2} \) |
| 7 | \( 1 + (-0.0601 + 0.340i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (0.377 - 0.137i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.575 - 0.483i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.670 - 1.16i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.87 + 3.24i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.536 + 3.04i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.79 - 3.18i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.774 + 4.39i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.25 - 2.16i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.73 + 1.45i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (8.03 - 2.92i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (2.13 - 12.1i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + (-12.7 - 4.64i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-2.31 + 13.1i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (9.40 - 7.89i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.14 + 1.98i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (6.23 - 10.8i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-9.11 - 7.64i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-3.71 - 3.11i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (0.197 - 0.341i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.9 + 5.07i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.28624659272796074190820862780, −9.404947062562740564403581215558, −8.416240964278201662794907987073, −7.46615187096169703349550078335, −6.77522060470762273242906885647, −6.00838462334615200360175050166, −4.99142798520199354980952317442, −3.89763003808994751971578013375, −2.60199679454719406380133663904, −1.05491623349453128584434891938,
2.12599159912207851973369320870, 3.20270766564082093720628363547, 3.91853340170749070122896833616, 4.96359386071825799315445755358, 5.64743458532233604246053713054, 7.30352325256182788303050322806, 8.205085294431949057596639781252, 8.913684471919943670396581290230, 10.06339899104003695592990151405, 10.53898735963651999036162144577
