| L(s) = 1 | + (0.610 − 0.132i)3-s + (0.110 − 2.23i)5-s + (4.16 − 2.27i)7-s + (−2.37 + 1.08i)9-s + (−4.03 − 3.49i)11-s + (0.486 − 0.890i)13-s + (−0.228 − 1.37i)15-s + (−0.383 − 0.287i)17-s + (−0.949 + 6.60i)19-s + (2.24 − 1.94i)21-s + (2.51 − 4.08i)23-s + (−4.97 − 0.495i)25-s + (−2.80 + 2.10i)27-s + (6.66 − 0.958i)29-s + (3.69 − 2.37i)31-s + ⋯ |
| L(s) = 1 | + (0.352 − 0.0766i)3-s + (0.0495 − 0.998i)5-s + (1.57 − 0.859i)7-s + (−0.791 + 0.361i)9-s + (−1.21 − 1.05i)11-s + (0.134 − 0.247i)13-s + (−0.0590 − 0.355i)15-s + (−0.0930 − 0.0696i)17-s + (−0.217 + 1.51i)19-s + (0.488 − 0.423i)21-s + (0.524 − 0.851i)23-s + (−0.995 − 0.0990i)25-s + (−0.539 + 0.404i)27-s + (1.23 − 0.178i)29-s + (0.664 − 0.426i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.259 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.259 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27101 - 0.974478i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27101 - 0.974478i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.110 + 2.23i)T \) |
| 23 | \( 1 + (-2.51 + 4.08i)T \) |
| good | 3 | \( 1 + (-0.610 + 0.132i)T + (2.72 - 1.24i)T^{2} \) |
| 7 | \( 1 + (-4.16 + 2.27i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (4.03 + 3.49i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.486 + 0.890i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (0.383 + 0.287i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (0.949 - 6.60i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-6.66 + 0.958i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-3.69 + 2.37i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-2.66 - 0.992i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (-3.53 + 7.73i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-2.58 - 11.8i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (4.04 + 4.04i)T + 47iT^{2} \) |
| 53 | \( 1 + (-3.38 - 6.20i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (-2.34 - 7.97i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-0.276 - 0.430i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-4.79 - 0.343i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (-5.89 - 6.79i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (0.994 + 1.32i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (11.8 - 3.47i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (2.40 - 6.45i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (-0.590 - 0.379i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (0.420 + 1.12i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−10.88425891075638302908549228067, −10.15510786141709972019864875226, −8.586199975319382548646519658531, −8.241561603770054506929916671197, −7.70359448203064291671279399934, −5.89517532155944871062884314290, −5.09918555707902208432274001437, −4.16518683766975963389339673317, −2.56538150406063971364083215540, −1.01348823130488788952447151380,
2.17981561090912218635853264430, 2.89734016493205117838203922050, 4.63529265902485652168737817684, 5.43208718211373083552474122884, 6.70818988281565461153703201473, 7.72029469010566535024142948744, 8.457635269762768549219614932405, 9.374553836545883482887856941280, 10.48907242832959829950350281717, 11.30885212784032061833484950006
