| L(s) = 1 | + (−0.689 − 0.0750i)2-s + (−1.94 + 2.28i)3-s + (−7.34 − 1.61i)4-s + (−13.2 − 10.0i)5-s + (1.51 − 1.43i)6-s + (3.00 − 7.53i)7-s + (10.2 + 3.43i)8-s + (−1.45 − 8.88i)9-s + (8.38 + 7.94i)10-s + (−20.9 + 9.69i)11-s + (17.9 − 13.6i)12-s + (0.115 − 0.704i)13-s + (−2.63 + 4.97i)14-s + (48.7 − 10.7i)15-s + (47.8 + 22.1i)16-s + (21.2 + 53.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.243 − 0.0265i)2-s + (−0.373 + 0.440i)3-s + (−0.917 − 0.202i)4-s + (−1.18 − 0.901i)5-s + (0.102 − 0.0974i)6-s + (0.162 − 0.406i)7-s + (0.450 + 0.151i)8-s + (−0.0539 − 0.328i)9-s + (0.265 + 0.251i)10-s + (−0.574 + 0.265i)11-s + (0.431 − 0.328i)12-s + (0.00246 − 0.0150i)13-s + (−0.0503 + 0.0949i)14-s + (0.839 − 0.184i)15-s + (0.747 + 0.345i)16-s + (0.303 + 0.761i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.482561 + 0.236026i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.482561 + 0.236026i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.94 - 2.28i)T \) |
| 59 | \( 1 + (165. + 421. i)T \) |
| good | 2 | \( 1 + (0.689 + 0.0750i)T + (7.81 + 1.71i)T^{2} \) |
| 5 | \( 1 + (13.2 + 10.0i)T + (33.4 + 120. i)T^{2} \) |
| 7 | \( 1 + (-3.00 + 7.53i)T + (-249. - 235. i)T^{2} \) |
| 11 | \( 1 + (20.9 - 9.69i)T + (861. - 1.01e3i)T^{2} \) |
| 13 | \( 1 + (-0.115 + 0.704i)T + (-2.08e3 - 701. i)T^{2} \) |
| 17 | \( 1 + (-21.2 - 53.3i)T + (-3.56e3 + 3.37e3i)T^{2} \) |
| 19 | \( 1 + (-13.2 - 19.4i)T + (-2.53e3 + 6.37e3i)T^{2} \) |
| 23 | \( 1 + (1.25 - 23.1i)T + (-1.20e4 - 1.31e3i)T^{2} \) |
| 29 | \( 1 + (52.4 - 5.70i)T + (2.38e4 - 5.24e3i)T^{2} \) |
| 31 | \( 1 + (16.3 - 24.1i)T + (-1.10e4 - 2.76e4i)T^{2} \) |
| 37 | \( 1 + (-275. + 92.9i)T + (4.03e4 - 3.06e4i)T^{2} \) |
| 41 | \( 1 + (-6.08 - 112. i)T + (-6.85e4 + 7.45e3i)T^{2} \) |
| 43 | \( 1 + (-218. - 101. i)T + (5.14e4 + 6.05e4i)T^{2} \) |
| 47 | \( 1 + (-113. + 86.0i)T + (2.77e4 - 1.00e5i)T^{2} \) |
| 53 | \( 1 + (-28.2 + 26.7i)T + (8.06e3 - 1.48e5i)T^{2} \) |
| 61 | \( 1 + (-694. - 75.4i)T + (2.21e5 + 4.87e4i)T^{2} \) |
| 67 | \( 1 + (349. + 117. i)T + (2.39e5 + 1.82e5i)T^{2} \) |
| 71 | \( 1 + (789. - 600. i)T + (9.57e4 - 3.44e5i)T^{2} \) |
| 73 | \( 1 + (-209. + 395. i)T + (-2.18e5 - 3.21e5i)T^{2} \) |
| 79 | \( 1 + (-350. - 412. i)T + (-7.97e4 + 4.86e5i)T^{2} \) |
| 83 | \( 1 + (813. - 489. i)T + (2.67e5 - 5.05e5i)T^{2} \) |
| 89 | \( 1 + (1.03e3 - 112. i)T + (6.88e5 - 1.51e5i)T^{2} \) |
| 97 | \( 1 + (-561. - 1.05e3i)T + (-5.12e5 + 7.55e5i)T^{2} \) |
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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
−12.44799337456712421355200135605, −11.30667402584227781247242143060, −10.34598169909615670317201526505, −9.343035278397965494238670407847, −8.311060488797809000233468577050, −7.59119968804084967073559834346, −5.62427537740666537035403795428, −4.57635593588989136658722975321, −3.85179378016754670718752883709, −0.903963072373797095554422738301,
0.41455166089343522390827172643, 2.93074662456969510154121403041, 4.30776387137562520754999379457, 5.61887344010310563041185714408, 7.20506908075782791928377241521, 7.83394912445641381823085902445, 8.869564134690523140968165152762, 10.19626412008998090794108053736, 11.22887917239258316727082167143, 11.98015349953475643531931094686
