| L(s) = 1 | + (0.347 + 1.37i)2-s + (0.139 + 0.0578i)3-s + (−1.75 + 0.953i)4-s + (2.91 − 1.48i)5-s + (−0.0307 + 0.211i)6-s + (2.40 + 3.92i)7-s + (−1.91 − 2.07i)8-s + (−2.10 − 2.10i)9-s + (3.04 + 3.47i)10-s + (−2.31 − 0.182i)11-s + (−0.300 + 0.0313i)12-s + (0.743 + 3.09i)13-s + (−4.53 + 4.65i)14-s + (0.492 − 0.0387i)15-s + (2.18 − 3.35i)16-s + (0.899 − 0.768i)17-s + ⋯ |
| L(s) = 1 | + (0.245 + 0.969i)2-s + (0.0805 + 0.0333i)3-s + (−0.879 + 0.476i)4-s + (1.30 − 0.663i)5-s + (−0.0125 + 0.0863i)6-s + (0.908 + 1.48i)7-s + (−0.678 − 0.735i)8-s + (−0.701 − 0.701i)9-s + (0.963 + 1.09i)10-s + (−0.697 − 0.0549i)11-s + (−0.0867 + 0.00905i)12-s + (0.206 + 0.858i)13-s + (−1.21 + 1.24i)14-s + (0.127 − 0.0100i)15-s + (0.545 − 0.837i)16-s + (0.218 − 0.186i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.222 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.222 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.10950 + 0.884724i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10950 + 0.884724i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.347 - 1.37i)T \) |
| 41 | \( 1 + (-6.34 + 0.889i)T \) |
| good | 3 | \( 1 + (-0.139 - 0.0578i)T + (2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (-2.91 + 1.48i)T + (2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (-2.40 - 3.92i)T + (-3.17 + 6.23i)T^{2} \) |
| 11 | \( 1 + (2.31 + 0.182i)T + (10.8 + 1.72i)T^{2} \) |
| 13 | \( 1 + (-0.743 - 3.09i)T + (-11.5 + 5.90i)T^{2} \) |
| 17 | \( 1 + (-0.899 + 0.768i)T + (2.65 - 16.7i)T^{2} \) |
| 19 | \( 1 + (5.22 + 1.25i)T + (16.9 + 8.62i)T^{2} \) |
| 23 | \( 1 + (-4.43 + 3.22i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (3.36 + 2.87i)T + (4.53 + 28.6i)T^{2} \) |
| 31 | \( 1 + (1.90 + 5.87i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.773 + 2.38i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (-0.766 - 0.121i)T + (40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (3.64 - 5.94i)T + (-21.3 - 41.8i)T^{2} \) |
| 53 | \( 1 + (3.29 - 3.85i)T + (-8.29 - 52.3i)T^{2} \) |
| 59 | \( 1 + (5.52 + 7.60i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (0.739 - 0.117i)T + (58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (0.809 + 10.2i)T + (-66.1 + 10.4i)T^{2} \) |
| 71 | \( 1 + (-0.0829 + 1.05i)T + (-70.1 - 11.1i)T^{2} \) |
| 73 | \( 1 + (8.87 - 8.87i)T - 73iT^{2} \) |
| 79 | \( 1 + (4.61 - 11.1i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 - 7.38iT - 83T^{2} \) |
| 89 | \( 1 + (-11.7 + 7.21i)T + (40.4 - 79.2i)T^{2} \) |
| 97 | \( 1 + (-1.48 - 18.8i)T + (-95.8 + 15.1i)T^{2} \) |
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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
−13.11370392606115995157431242067, −12.43568211987880682512767142800, −11.21813091047763097704102069136, −9.321971786682715619276584790043, −9.041746649648577427927436686013, −8.078633916919624585175131248471, −6.26502400299792563175402667162, −5.65688445084703745087154660547, −4.67803111411630725014257673507, −2.40039599180040254391737898013,
1.74956912359156454259850566221, 3.15417747882430034551388461929, 4.84302659592085406894768850737, 5.82972738781978155193417283884, 7.50902465892696383246708904762, 8.678177087630874029509493149646, 10.22450674748445418690726453989, 10.57521536427272179734245394708, 11.21321651628159347972354134093, 13.01408356126104749675012306454
