L(s) = 1 | + (−1.33 − 2.31i)5-s − 3.93i·7-s + 2.20i·11-s + (−3.60 − 2.07i)13-s + (0.571 + 0.990i)17-s + (−4.24 − 0.990i)19-s + (−3.19 − 1.84i)23-s + (−1.07 + 1.85i)25-s + (2.10 + 1.21i)29-s + 3.67·31-s + (−9.11 + 5.25i)35-s − 10.0i·37-s + (−8.01 + 4.62i)41-s + (−0.490 + 0.283i)43-s + (10.6 + 6.13i)47-s + ⋯ |
L(s) = 1 | + (−0.597 − 1.03i)5-s − 1.48i·7-s + 0.664i·11-s + (−0.998 − 0.576i)13-s + (0.138 + 0.240i)17-s + (−0.973 − 0.227i)19-s + (−0.665 − 0.384i)23-s + (−0.214 + 0.371i)25-s + (0.390 + 0.225i)29-s + 0.659·31-s + (−1.53 + 0.889i)35-s − 1.65i·37-s + (−1.25 + 0.723i)41-s + (−0.0748 + 0.0432i)43-s + (1.55 + 0.895i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3245949102\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3245949102\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (4.24 + 0.990i)T \) |
good | 5 | \( 1 + (1.33 + 2.31i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + 3.93iT - 7T^{2} \) |
| 11 | \( 1 - 2.20iT - 11T^{2} \) |
| 13 | \( 1 + (3.60 + 2.07i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.571 - 0.990i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (3.19 + 1.84i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.10 - 1.21i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 3.67T + 31T^{2} \) |
| 37 | \( 1 + 10.0iT - 37T^{2} \) |
| 41 | \( 1 + (8.01 - 4.62i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.490 - 0.283i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-10.6 - 6.13i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.09 + 2.36i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.33 - 2.31i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.41 - 11.1i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.73 + 3.00i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.24 - 10.8i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.35 - 2.34i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.50 + 6.07i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 1.98iT - 83T^{2} \) |
| 89 | \( 1 + (12.9 + 7.46i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.91 + 1.68i)T + (48.5 - 84.0i)T^{2} \) |
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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
−8.271194178949169613242357295585, −7.56671339181505091901446265785, −7.08438595805300849670679730585, −6.05734840585122605060114220358, −4.87184100134077535714011002600, −4.44206956973426094927649550065, −3.78420691725536498431461479310, −2.44145995979417562704916976541, −1.11335211582191243568453988943, −0.11304189279384620824298217223,
2.00563860365815473217296616193, 2.76083049711923952904135286663, 3.53664552375241879507488428782, 4.64300864439135456158848312423, 5.51424110527561760730537980899, 6.37098770431608098996524586919, 6.89072553832300704829028480012, 7.921632469426460224985709784840, 8.449671656573502996824975318424, 9.257053790799120260198837649158