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
−13.49543092696308834599980227813, −12.55727333000353373173184078432, −11.80609229930277681898788521658, −10.47971359069972218594769724566, −9.507530342374556887019535504105, −7.34721202496072338003420935016, −6.48480194431039499996738598409, −6.00008585878220079475870032806, −3.81897643489107998935708767703, −1.85733840359798182619057008653,
3.12033785201426551237037516639, 4.64771059124356936521190025623, 5.72217224103226655613412412811, 6.50223488126118898979469575372, 8.863649532677348627373388108610, 9.672544116850437866238243995466, 10.64062247643625754395556455905, 11.85823261494576192061262481617, 12.85270546443869549593141196928, 13.73948119716483529350826065006