| L(s) = 1 | + (0.763 + 1.19i)2-s + (−2.62 − 0.258i)3-s + (−0.835 + 1.81i)4-s + (−0.471 − 0.881i)5-s + (−1.69 − 3.32i)6-s + (0.894 + 4.49i)7-s + (−2.80 + 0.392i)8-s + (3.87 + 0.770i)9-s + (0.690 − 1.23i)10-s + (−2.43 − 1.99i)11-s + (2.66 − 4.55i)12-s + (0.202 + 0.108i)13-s + (−4.67 + 4.49i)14-s + (1.00 + 2.43i)15-s + (−2.60 − 3.03i)16-s + (0.635 − 1.53i)17-s + ⋯ |
| L(s) = 1 | + (0.539 + 0.841i)2-s + (−1.51 − 0.149i)3-s + (−0.417 + 0.908i)4-s + (−0.210 − 0.394i)5-s + (−0.691 − 1.35i)6-s + (0.338 + 1.69i)7-s + (−0.990 + 0.138i)8-s + (1.29 + 0.256i)9-s + (0.218 − 0.390i)10-s + (−0.733 − 0.602i)11-s + (0.768 − 1.31i)12-s + (0.0561 + 0.0299i)13-s + (−1.24 + 1.20i)14-s + (0.260 + 0.628i)15-s + (−0.651 − 0.758i)16-s + (0.154 − 0.372i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0842 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0842 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00520210 - 0.00566064i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00520210 - 0.00566064i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.763 - 1.19i)T \) |
| 5 | \( 1 + (0.471 + 0.881i)T \) |
| good | 3 | \( 1 + (2.62 + 0.258i)T + (2.94 + 0.585i)T^{2} \) |
| 7 | \( 1 + (-0.894 - 4.49i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (2.43 + 1.99i)T + (2.14 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-0.202 - 0.108i)T + (7.22 + 10.8i)T^{2} \) |
| 17 | \( 1 + (-0.635 + 1.53i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (6.29 - 1.90i)T + (15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (-1.91 + 1.28i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (4.93 + 6.00i)T + (-5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (-3.13 + 3.13i)T - 31iT^{2} \) |
| 37 | \( 1 + (-2.62 + 8.65i)T + (-30.7 - 20.5i)T^{2} \) |
| 41 | \( 1 + (2.50 + 3.75i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-5.63 + 0.555i)T + (42.1 - 8.38i)T^{2} \) |
| 47 | \( 1 + (11.0 + 4.55i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (0.888 - 1.08i)T + (-10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (8.90 - 4.75i)T + (32.7 - 49.0i)T^{2} \) |
| 61 | \( 1 + (1.38 - 14.0i)T + (-59.8 - 11.9i)T^{2} \) |
| 67 | \( 1 + (1.13 - 11.5i)T + (-65.7 - 13.0i)T^{2} \) |
| 71 | \( 1 + (9.07 - 1.80i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-0.122 + 0.617i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-9.87 + 4.09i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (1.51 + 4.99i)T + (-69.0 + 46.1i)T^{2} \) |
| 89 | \( 1 + (-4.70 - 3.14i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (4.23 - 4.23i)T - 97iT^{2} \) |
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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
−11.59119613894774323622914646365, −10.63122500005443655971979475259, −9.198389002552830908074323198556, −8.452709949418369920766182117761, −7.59236538206461487322823313383, −6.28157953584972957845383059383, −5.78049209528934856843265001458, −5.21700697555162349437063442595, −4.24837621171457179384578011230, −2.48825865489620347636906253081,
0.00429957487004709306464814236, 1.47251951726819529373141458868, 3.35552355735028783985605540308, 4.59836631846377102104693868285, 4.85548901879705145742574799875, 6.26914925953234999416793105040, 6.88193737319993843017070385041, 8.040340181218973257099889709672, 9.703193898243528487565877417116, 10.43336084888292195956458526014
