| L(s) = 1 | + (−0.415 − 0.909i)3-s + (0.182 + 0.210i)5-s + (1.45 + 0.932i)7-s + (−0.654 + 0.755i)9-s + (0.807 + 5.61i)11-s + (−0.0913 + 0.0587i)13-s + (0.115 − 0.253i)15-s + (2.82 − 0.829i)17-s + (0.588 + 0.172i)19-s + (0.245 − 1.70i)21-s + (3.11 + 3.64i)23-s + (0.700 − 4.87i)25-s + (0.959 + 0.281i)27-s + (1.69 − 0.497i)29-s + (0.0756 − 0.165i)31-s + ⋯ |
| L(s) = 1 | + (−0.239 − 0.525i)3-s + (0.0815 + 0.0941i)5-s + (0.548 + 0.352i)7-s + (−0.218 + 0.251i)9-s + (0.243 + 1.69i)11-s + (−0.0253 + 0.0162i)13-s + (0.0298 − 0.0653i)15-s + (0.684 − 0.201i)17-s + (0.135 + 0.0396i)19-s + (0.0535 − 0.372i)21-s + (0.650 + 0.759i)23-s + (0.140 − 0.974i)25-s + (0.184 + 0.0542i)27-s + (0.314 − 0.0923i)29-s + (0.0135 − 0.0297i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.42978 + 0.228080i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42978 + 0.228080i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.415 + 0.909i)T \) |
| 23 | \( 1 + (-3.11 - 3.64i)T \) |
| good | 5 | \( 1 + (-0.182 - 0.210i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (-1.45 - 0.932i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.807 - 5.61i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (0.0913 - 0.0587i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-2.82 + 0.829i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-0.588 - 0.172i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-1.69 + 0.497i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.0756 + 0.165i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-2.35 + 2.72i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-2.18 - 2.52i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-3.35 - 7.35i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 0.555T + 47T^{2} \) |
| 53 | \( 1 + (-8.03 - 5.16i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (8.23 - 5.29i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (0.481 - 1.05i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-1.56 + 10.8i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.34 + 9.34i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (5.65 + 1.65i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-3.05 + 1.96i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (10.0 - 11.6i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (3.39 + 7.43i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (8.41 + 9.71i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−10.89827391584127238651246275493, −9.919607786274485385527878536036, −9.151031562325366467553394979485, −7.929343571232564883472369459583, −7.31851242098939270488379478660, −6.33197162096000225359851269487, −5.23903575352291123689179915441, −4.35008063305989554730614611157, −2.66979788920820895793830871010, −1.47762255771138010980978392317,
1.01469531631719669496336602198, 3.01456448198235168358131002039, 4.03071698278302505003929149332, 5.21016655132116371767178127545, 5.95565791207558479015308100814, 7.12236253469536470029277686396, 8.285188403753047481246921428165, 8.896555250133969909685907103203, 9.963646850751383677753738044476, 10.90436692588692457251020244793
