| L(s) = 1 | + (0.973 − 0.230i)2-s + (−1.71 − 0.215i)3-s + (0.893 − 0.448i)4-s + (−1.74 − 2.33i)5-s + (−1.72 + 0.186i)6-s + (3.14 − 2.06i)7-s + (0.766 − 0.642i)8-s + (2.90 + 0.740i)9-s + (−2.23 − 1.87i)10-s + (−2.28 − 0.267i)11-s + (−1.63 + 0.578i)12-s + (−1.19 − 4.00i)13-s + (2.58 − 2.73i)14-s + (2.48 + 4.39i)15-s + (0.597 − 0.802i)16-s + (−0.456 + 2.58i)17-s + ⋯ |
| L(s) = 1 | + (0.688 − 0.163i)2-s + (−0.992 − 0.124i)3-s + (0.446 − 0.224i)4-s + (−0.778 − 1.04i)5-s + (−0.702 + 0.0761i)6-s + (1.18 − 0.781i)7-s + (0.270 − 0.227i)8-s + (0.969 + 0.246i)9-s + (−0.706 − 0.592i)10-s + (−0.688 − 0.0805i)11-s + (−0.471 + 0.167i)12-s + (−0.332 − 1.11i)13-s + (0.689 − 0.731i)14-s + (0.642 + 1.13i)15-s + (0.149 − 0.200i)16-s + (−0.110 + 0.627i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.216 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.930877 - 0.746940i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.930877 - 0.746940i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.973 + 0.230i)T \) |
| 3 | \( 1 + (1.71 + 0.215i)T \) |
| good | 5 | \( 1 + (1.74 + 2.33i)T + (-1.43 + 4.78i)T^{2} \) |
| 7 | \( 1 + (-3.14 + 2.06i)T + (2.77 - 6.42i)T^{2} \) |
| 11 | \( 1 + (2.28 + 0.267i)T + (10.7 + 2.53i)T^{2} \) |
| 13 | \( 1 + (1.19 + 4.00i)T + (-10.8 + 7.14i)T^{2} \) |
| 17 | \( 1 + (0.456 - 2.58i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (-1.50 - 8.52i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (-6.27 - 4.12i)T + (9.10 + 21.1i)T^{2} \) |
| 29 | \( 1 + (1.68 + 1.78i)T + (-1.68 + 28.9i)T^{2} \) |
| 31 | \( 1 + (-0.165 + 2.83i)T + (-30.7 - 3.59i)T^{2} \) |
| 37 | \( 1 + (-5.99 + 2.18i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (4.24 + 1.00i)T + (36.6 + 18.4i)T^{2} \) |
| 43 | \( 1 + (-4.40 - 10.2i)T + (-29.5 + 31.2i)T^{2} \) |
| 47 | \( 1 + (0.0956 + 1.64i)T + (-46.6 + 5.45i)T^{2} \) |
| 53 | \( 1 + (-0.561 + 0.972i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-9.21 + 1.07i)T + (57.4 - 13.6i)T^{2} \) |
| 61 | \( 1 + (3.19 + 1.60i)T + (36.4 + 48.9i)T^{2} \) |
| 67 | \( 1 + (1.00 - 1.06i)T + (-3.89 - 66.8i)T^{2} \) |
| 71 | \( 1 + (1.46 + 1.23i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (0.123 - 0.103i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (9.27 - 2.19i)T + (70.5 - 35.4i)T^{2} \) |
| 83 | \( 1 + (2.15 - 0.511i)T + (74.1 - 37.2i)T^{2} \) |
| 89 | \( 1 + (0.717 - 0.602i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (3.02 - 4.06i)T + (-27.8 - 92.9i)T^{2} \) |
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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
−12.68686641113689978347193913721, −11.71168608674235737404514420521, −10.97844463149666491078055764542, −10.06477272328618982220953521774, −8.011484888996434422391972204573, −7.59512508415524848142243137827, −5.71686381559324483233541698968, −4.94345215371640811203453197014, −3.96496032254379329392350364019, −1.20014358793692803879988970430,
2.65158405595054945166083088855, 4.54269752351363815232743653120, 5.21777970513390451295662984595, 6.81778773822807179519211879264, 7.33598304674275184441256012225, 8.941243991489240866030483876750, 10.65010321557642196340554542785, 11.43496249194708033153158065202, 11.68759158916336877423047967379, 12.94619134035025979823984520285
