| L(s) = 1 | + (0.193 + 1.22i)2-s + (−2.26 + 1.15i)3-s + (0.443 − 0.144i)4-s + (−0.536 + 2.17i)5-s + (−1.85 − 2.54i)6-s + (1.57 − 3.09i)7-s + (1.38 + 2.72i)8-s + (2.04 − 2.80i)9-s + (−2.75 − 0.235i)10-s + (0.937 − 3.18i)11-s + (−0.839 + 0.839i)12-s + (0.730 − 0.115i)13-s + (4.08 + 1.32i)14-s + (−1.29 − 5.54i)15-s + (−2.30 + 1.67i)16-s + (−0.609 − 0.0965i)17-s + ⋯ |
| L(s) = 1 | + (0.136 + 0.864i)2-s + (−1.30 + 0.666i)3-s + (0.221 − 0.0720i)4-s + (−0.239 + 0.970i)5-s + (−0.755 − 1.04i)6-s + (0.595 − 1.16i)7-s + (0.490 + 0.962i)8-s + (0.680 − 0.936i)9-s + (−0.872 − 0.0744i)10-s + (0.282 − 0.959i)11-s + (−0.242 + 0.242i)12-s + (0.202 − 0.0320i)13-s + (1.09 + 0.354i)14-s + (−0.333 − 1.43i)15-s + (−0.576 + 0.418i)16-s + (−0.147 − 0.0234i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.490778 + 0.545097i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.490778 + 0.545097i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.536 - 2.17i)T \) |
| 11 | \( 1 + (-0.937 + 3.18i)T \) |
| good | 2 | \( 1 + (-0.193 - 1.22i)T + (-1.90 + 0.618i)T^{2} \) |
| 3 | \( 1 + (2.26 - 1.15i)T + (1.76 - 2.42i)T^{2} \) |
| 7 | \( 1 + (-1.57 + 3.09i)T + (-4.11 - 5.66i)T^{2} \) |
| 13 | \( 1 + (-0.730 + 0.115i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (0.609 + 0.0965i)T + (16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (0.971 - 2.99i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (4.30 + 4.30i)T + 23iT^{2} \) |
| 29 | \( 1 + (-0.896 - 2.75i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (2.45 + 1.78i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.04 - 1.04i)T + (21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (-0.970 - 0.315i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-4.07 + 4.07i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.492 + 0.967i)T + (-27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (0.671 + 4.24i)T + (-50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (7.03 - 2.28i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (2.20 + 3.03i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (9.39 - 9.39i)T - 67iT^{2} \) |
| 71 | \( 1 + (2.92 - 2.12i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (0.479 + 0.244i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (-9.87 - 7.17i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (2.50 - 15.8i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + 14.6iT - 89T^{2} \) |
| 97 | \( 1 + (-14.1 + 2.24i)T + (92.2 - 29.9i)T^{2} \) |
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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
−15.89259832763877706051763158298, −14.65413308887404888315656224707, −13.92220949713260417919998689891, −11.74948806620884617055424470831, −10.86936945921835702540106847633, −10.44607456939842040682535768156, −7.972168957312376680585780671014, −6.70492514948441848543379614814, −5.78679632060962614644794923866, −4.19362398092889181747246390347,
1.75439518332264450609609235534, 4.65390929457750357430755361487, 6.03423753603751555224832820628, 7.58517741399089727199820945541, 9.307876682657034053209906588818, 10.98083249162101526466197397520, 11.95663211049707511257133762512, 12.20494485567282101624870350723, 13.22053364856700552030768652736, 15.30024077763471424241189454443
