| L(s) = 1 | + (−1.68 + 0.222i)2-s + (1.30 + 1.13i)3-s + (0.871 − 0.233i)4-s + (−0.431 − 0.875i)5-s + (−2.46 − 1.62i)6-s + (2.92 + 1.44i)7-s + (1.72 − 0.715i)8-s + (0.429 + 2.96i)9-s + (0.923 + 1.38i)10-s + (−3.22 + 1.09i)11-s + (1.40 + 0.681i)12-s + (0.485 + 1.81i)13-s + (−5.25 − 1.78i)14-s + (0.427 − 1.63i)15-s + (−4.32 + 2.49i)16-s + (3.35 + 2.39i)17-s + ⋯ |
| L(s) = 1 | + (−1.19 + 0.157i)2-s + (0.756 + 0.654i)3-s + (0.435 − 0.116i)4-s + (−0.193 − 0.391i)5-s + (−1.00 − 0.662i)6-s + (1.10 + 0.544i)7-s + (0.611 − 0.253i)8-s + (0.143 + 0.989i)9-s + (0.292 + 0.437i)10-s + (−0.972 + 0.330i)11-s + (0.405 + 0.196i)12-s + (0.134 + 0.502i)13-s + (−1.40 − 0.477i)14-s + (0.110 − 0.422i)15-s + (−1.08 + 0.623i)16-s + (0.813 + 0.581i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.665926 + 0.430111i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.665926 + 0.430111i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.30 - 1.13i)T \) |
| 17 | \( 1 + (-3.35 - 2.39i)T \) |
| good | 2 | \( 1 + (1.68 - 0.222i)T + (1.93 - 0.517i)T^{2} \) |
| 5 | \( 1 + (0.431 + 0.875i)T + (-3.04 + 3.96i)T^{2} \) |
| 7 | \( 1 + (-2.92 - 1.44i)T + (4.26 + 5.55i)T^{2} \) |
| 11 | \( 1 + (3.22 - 1.09i)T + (8.72 - 6.69i)T^{2} \) |
| 13 | \( 1 + (-0.485 - 1.81i)T + (-11.2 + 6.5i)T^{2} \) |
| 19 | \( 1 + (-1.65 - 0.683i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (2.05 - 2.34i)T + (-3.00 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.427 + 6.52i)T + (-28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (-0.365 + 1.07i)T + (-24.5 - 18.8i)T^{2} \) |
| 37 | \( 1 + (1.75 + 8.83i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-4.80 + 0.315i)T + (40.6 - 5.35i)T^{2} \) |
| 43 | \( 1 + (8.86 + 6.80i)T + (11.1 + 41.5i)T^{2} \) |
| 47 | \( 1 + (-0.942 + 3.51i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.730 + 1.76i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (0.828 - 6.29i)T + (-56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (2.95 - 5.99i)T + (-37.1 - 48.3i)T^{2} \) |
| 67 | \( 1 + (-4.41 - 2.54i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-16.3 + 3.25i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (13.5 + 9.05i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (0.357 + 1.05i)T + (-62.6 + 48.0i)T^{2} \) |
| 83 | \( 1 + (1.10 + 8.40i)T + (-80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (9.18 - 9.18i)T - 89iT^{2} \) |
| 97 | \( 1 + (-13.2 - 0.866i)T + (96.1 + 12.6i)T^{2} \) |
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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
−13.30674450503835379075529518657, −11.94305463015756239521838333817, −10.70546089484927418096826548127, −9.926300032581000751269003388869, −8.902954392037994588539651749845, −8.161629806648616497993499541832, −7.57535793190324316315783726894, −5.32221450379896763395084673489, −4.16269217607602822077789214540, −2.01637801268566884280472913577,
1.25575115812479920572508084347, 3.01003921018851214084654785520, 5.00208952293479389386924380376, 7.03621133752963784296983004009, 7.922505115297639775785224293684, 8.344629891199583070098242592563, 9.661846187867697126312913302842, 10.63676336440179098288619153273, 11.46277639110268613330596049102, 12.86166515721141997299175643611
