| 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
−12.94619134035025979823984520285, −11.68759158916336877423047967379, −11.43496249194708033153158065202, −10.65010321557642196340554542785, −8.941243991489240866030483876750, −7.33598304674275184441256012225, −6.81778773822807179519211879264, −5.21777970513390451295662984595, −4.54269752351363815232743653120, −2.65158405595054945166083088855,
1.20014358793692803879988970430, 3.96496032254379329392350364019, 4.94345215371640811203453197014, 5.71686381559324483233541698968, 7.59512508415524848142243137827, 8.011484888996434422391972204573, 10.06477272328618982220953521774, 10.97844463149666491078055764542, 11.71168608674235737404514420521, 12.68686641113689978347193913721
