L(s) = 1 | + (1.39 + 0.250i)2-s + (1.01 − 1.40i)3-s + (1.87 + 0.696i)4-s + (−2.23 − 0.116i)5-s + (1.76 − 1.69i)6-s + (−2.29 + 2.29i)7-s + (2.43 + 1.43i)8-s + (−0.925 − 2.85i)9-s + (−3.07 − 0.720i)10-s − 2.28·11-s + (2.88 − 1.91i)12-s + (1.05 − 1.05i)13-s + (−3.76 + 2.61i)14-s + (−2.43 + 3.00i)15-s + (3.03 + 2.61i)16-s + (3.04 + 3.04i)17-s + ⋯ |
L(s) = 1 | + (0.984 + 0.176i)2-s + (0.588 − 0.808i)3-s + (0.937 + 0.348i)4-s + (−0.998 − 0.0519i)5-s + (0.721 − 0.692i)6-s + (−0.865 + 0.865i)7-s + (0.861 + 0.508i)8-s + (−0.308 − 0.951i)9-s + (−0.973 − 0.227i)10-s − 0.688·11-s + (0.832 − 0.553i)12-s + (0.292 − 0.292i)13-s + (−1.00 + 0.698i)14-s + (−0.629 + 0.777i)15-s + (0.757 + 0.652i)16-s + (0.738 + 0.738i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.73329 - 0.170982i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.73329 - 0.170982i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.39 - 0.250i)T \) |
| 3 | \( 1 + (-1.01 + 1.40i)T \) |
| 5 | \( 1 + (2.23 + 0.116i)T \) |
good | 7 | \( 1 + (2.29 - 2.29i)T - 7iT^{2} \) |
| 11 | \( 1 + 2.28T + 11T^{2} \) |
| 13 | \( 1 + (-1.05 + 1.05i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3.04 - 3.04i)T + 17iT^{2} \) |
| 19 | \( 1 + 3.36T + 19T^{2} \) |
| 23 | \( 1 + (-3.68 + 3.68i)T - 23iT^{2} \) |
| 29 | \( 1 + 2.71iT - 29T^{2} \) |
| 31 | \( 1 + 6.49T + 31T^{2} \) |
| 37 | \( 1 + (-2.31 - 2.31i)T + 37iT^{2} \) |
| 41 | \( 1 + 10.8iT - 41T^{2} \) |
| 43 | \( 1 + (1.16 - 1.16i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.83 + 1.83i)T + 47iT^{2} \) |
| 53 | \( 1 + (-5.82 - 5.82i)T + 53iT^{2} \) |
| 59 | \( 1 - 7.41iT - 59T^{2} \) |
| 61 | \( 1 + 8.97iT - 61T^{2} \) |
| 67 | \( 1 + (-8.66 - 8.66i)T + 67iT^{2} \) |
| 71 | \( 1 - 7.37iT - 71T^{2} \) |
| 73 | \( 1 + (-1.83 - 1.83i)T + 73iT^{2} \) |
| 79 | \( 1 + 8.28iT - 79T^{2} \) |
| 83 | \( 1 + (5.27 + 5.27i)T + 83iT^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 + (2.79 - 2.79i)T - 97iT^{2} \) |
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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
−13.10833475658546002814765516394, −12.68611457031687562516319739614, −11.92567997588791401248898200788, −10.63643924654515772583486820642, −8.768900006681544833093619218259, −7.87092181815821649738201924799, −6.77583212803665820241976890697, −5.64412936629592742128139491280, −3.77338372295083905536486703812, −2.64982127043694289258862022966,
3.09713110015318820756830541090, 3.92096075349122477689781910497, 5.13278063036350406261352142616, 6.92617174673655988098345879685, 7.910558127086609728157796719407, 9.544880709057562103681074038640, 10.62398401980120016245846460735, 11.33953921043298250974351354973, 12.75230091495172076399976129348, 13.49952811692865906718761726837