| L(s) = 1 | + (0.186 − 0.139i)3-s + (1.92 + 1.13i)5-s + (−0.157 + 2.19i)7-s + (−0.829 + 2.82i)9-s + (−1.78 + 2.77i)11-s + (−3.55 + 0.253i)13-s + (0.517 − 0.0582i)15-s + (−1.34 + 3.60i)17-s + (3.52 − 7.71i)19-s + (0.277 + 0.431i)21-s + (3.03 − 3.71i)23-s + (2.43 + 4.36i)25-s + (0.484 + 1.29i)27-s + (−1.24 + 0.567i)29-s + (0.502 + 3.49i)31-s + ⋯ |
| L(s) = 1 | + (0.107 − 0.0806i)3-s + (0.862 + 0.506i)5-s + (−0.0593 + 0.829i)7-s + (−0.276 + 0.942i)9-s + (−0.537 + 0.836i)11-s + (−0.984 + 0.0704i)13-s + (0.133 − 0.0150i)15-s + (−0.326 + 0.874i)17-s + (0.808 − 1.76i)19-s + (0.0605 + 0.0941i)21-s + (0.631 − 0.775i)23-s + (0.487 + 0.872i)25-s + (0.0932 + 0.249i)27-s + (−0.230 + 0.105i)29-s + (0.0902 + 0.627i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16978 + 0.885776i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16978 + 0.885776i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.92 - 1.13i)T \) |
| 23 | \( 1 + (-3.03 + 3.71i)T \) |
| good | 3 | \( 1 + (-0.186 + 0.139i)T + (0.845 - 2.87i)T^{2} \) |
| 7 | \( 1 + (0.157 - 2.19i)T + (-6.92 - 0.996i)T^{2} \) |
| 11 | \( 1 + (1.78 - 2.77i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (3.55 - 0.253i)T + (12.8 - 1.85i)T^{2} \) |
| 17 | \( 1 + (1.34 - 3.60i)T + (-12.8 - 11.1i)T^{2} \) |
| 19 | \( 1 + (-3.52 + 7.71i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (1.24 - 0.567i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.502 - 3.49i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-2.52 - 4.62i)T + (-20.0 + 31.1i)T^{2} \) |
| 41 | \( 1 + (-7.52 + 2.20i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (3.12 + 4.17i)T + (-12.1 + 41.2i)T^{2} \) |
| 47 | \( 1 + (-4.59 + 4.59i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.64 + 0.189i)T + (52.4 + 7.54i)T^{2} \) |
| 59 | \( 1 + (-0.403 - 0.349i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-13.6 + 1.96i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (5.02 + 1.09i)T + (60.9 + 27.8i)T^{2} \) |
| 71 | \( 1 + (-5.32 + 3.42i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (4.96 - 1.85i)T + (55.1 - 47.8i)T^{2} \) |
| 79 | \( 1 + (-3.31 + 3.83i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-13.7 + 7.48i)T + (44.8 - 69.8i)T^{2} \) |
| 89 | \( 1 + (0.275 - 1.91i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (11.3 + 6.20i)T + (52.4 + 81.6i)T^{2} \) |
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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
−11.06299598308773013958579646309, −10.37601801801508974896221749573, −9.456123125454828585792215766172, −8.676606435438631049827446271589, −7.45281342479647695834553270092, −6.70039767816210612797140425308, −5.41383309986565558443454024779, −4.82432746302976429781024331563, −2.73589094999323992647490761550, −2.20371514578882929825198264640,
0.915247471309079026034990015772, 2.72476878482891117143100436556, 3.94827567208401852390434731798, 5.28647678302684338827426404928, 6.01831453908950913327083883319, 7.24515826345613726691540454795, 8.137113742089129729059947718208, 9.432768213218421284937355501591, 9.686051895836400752278354551840, 10.78444709546948196763237174449
