| L(s) = 1 | + (0.987 + 0.156i)2-s + (−1.07 + 2.10i)3-s + (0.951 + 0.309i)4-s + (1.91 − 1.15i)5-s + (−1.38 + 1.91i)6-s + (−3.37 + 1.71i)7-s + (0.891 + 0.453i)8-s + (−1.52 − 2.09i)9-s + (2.07 − 0.842i)10-s + (2.91 − 1.58i)11-s + (−1.67 + 1.67i)12-s + (0.378 − 2.38i)13-s + (−3.60 + 1.16i)14-s + (0.380 + 5.27i)15-s + (0.809 + 0.587i)16-s + (−0.299 − 1.89i)17-s + ⋯ |
| L(s) = 1 | + (0.698 + 0.110i)2-s + (−0.619 + 1.21i)3-s + (0.475 + 0.154i)4-s + (0.856 − 0.516i)5-s + (−0.567 + 0.780i)6-s + (−1.27 + 0.649i)7-s + (0.315 + 0.160i)8-s + (−0.507 − 0.698i)9-s + (0.655 − 0.266i)10-s + (0.877 − 0.479i)11-s + (−0.482 + 0.482i)12-s + (0.104 − 0.662i)13-s + (−0.962 + 0.312i)14-s + (0.0983 + 1.36i)15-s + (0.202 + 0.146i)16-s + (−0.0726 − 0.458i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.13635 + 0.643391i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13635 + 0.643391i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.987 - 0.156i)T \) |
| 5 | \( 1 + (-1.91 + 1.15i)T \) |
| 11 | \( 1 + (-2.91 + 1.58i)T \) |
| good | 3 | \( 1 + (1.07 - 2.10i)T + (-1.76 - 2.42i)T^{2} \) |
| 7 | \( 1 + (3.37 - 1.71i)T + (4.11 - 5.66i)T^{2} \) |
| 13 | \( 1 + (-0.378 + 2.38i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (0.299 + 1.89i)T + (-16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (1.20 + 3.70i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (2.12 + 2.12i)T + 23iT^{2} \) |
| 29 | \( 1 + (2.45 - 7.56i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.04 + 2.21i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.80 - 5.50i)T + (-21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (10.5 - 3.44i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-0.0971 + 0.0971i)T - 43iT^{2} \) |
| 47 | \( 1 + (-8.25 - 4.20i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-1.27 - 0.201i)T + (50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (13.8 + 4.50i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (2.01 - 2.77i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (7.14 - 7.14i)T - 67iT^{2} \) |
| 71 | \( 1 + (-10.0 - 7.32i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (4.54 + 8.92i)T + (-42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (-3.21 + 2.33i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (0.615 - 0.0974i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 - 2.93iT - 89T^{2} \) |
| 97 | \( 1 + (-0.904 + 5.70i)T + (-92.2 - 29.9i)T^{2} \) |
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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
−13.73948119716483529350826065006, −12.85270546443869549593141196928, −11.85823261494576192061262481617, −10.64062247643625754395556455905, −9.672544116850437866238243995466, −8.863649532677348627373388108610, −6.50223488126118898979469575372, −5.72217224103226655613412412811, −4.64771059124356936521190025623, −3.12033785201426551237037516639,
1.85733840359798182619057008653, 3.81897643489107998935708767703, 6.00008585878220079475870032806, 6.48480194431039499996738598409, 7.34721202496072338003420935016, 9.507530342374556887019535504105, 10.47971359069972218594769724566, 11.80609229930277681898788521658, 12.55727333000353373173184078432, 13.49543092696308834599980227813
