L(s) = 1 | + 2.51i·2-s + (−0.887 + 0.887i)3-s − 4.31·4-s + (0.707 − 0.707i)5-s + (−2.22 − 2.22i)6-s + (1.14 + 1.14i)7-s − 5.80i·8-s + 1.42i·9-s + (1.77 + 1.77i)10-s + (−2.32 − 2.32i)11-s + (3.82 − 3.82i)12-s + 6.35·13-s + (−2.86 + 2.86i)14-s + 1.25i·15-s + 5.96·16-s + (−0.768 + 4.05i)17-s + ⋯ |
L(s) = 1 | + 1.77i·2-s + (−0.512 + 0.512i)3-s − 2.15·4-s + (0.316 − 0.316i)5-s + (−0.909 − 0.909i)6-s + (0.431 + 0.431i)7-s − 2.05i·8-s + 0.475i·9-s + (0.561 + 0.561i)10-s + (−0.700 − 0.700i)11-s + (1.10 − 1.10i)12-s + 1.76·13-s + (−0.766 + 0.766i)14-s + 0.323i·15-s + 1.49·16-s + (−0.186 + 0.982i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.124877 + 0.799401i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.124877 + 0.799401i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.707 + 0.707i)T \) |
| 17 | \( 1 + (0.768 - 4.05i)T \) |
good | 2 | \( 1 - 2.51iT - 2T^{2} \) |
| 3 | \( 1 + (0.887 - 0.887i)T - 3iT^{2} \) |
| 7 | \( 1 + (-1.14 - 1.14i)T + 7iT^{2} \) |
| 11 | \( 1 + (2.32 + 2.32i)T + 11iT^{2} \) |
| 13 | \( 1 - 6.35T + 13T^{2} \) |
| 19 | \( 1 + 0.747iT - 19T^{2} \) |
| 23 | \( 1 + (-0.101 - 0.101i)T + 23iT^{2} \) |
| 29 | \( 1 + (-6.22 + 6.22i)T - 29iT^{2} \) |
| 31 | \( 1 + (-5.10 + 5.10i)T - 31iT^{2} \) |
| 37 | \( 1 + (0.439 - 0.439i)T - 37iT^{2} \) |
| 41 | \( 1 + (4.49 + 4.49i)T + 41iT^{2} \) |
| 43 | \( 1 - 2.74iT - 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 - 2.71iT - 53T^{2} \) |
| 59 | \( 1 - 12.6iT - 59T^{2} \) |
| 61 | \( 1 + (-0.328 - 0.328i)T + 61iT^{2} \) |
| 67 | \( 1 - 1.81T + 67T^{2} \) |
| 71 | \( 1 + (3.01 - 3.01i)T - 71iT^{2} \) |
| 73 | \( 1 + (0.856 - 0.856i)T - 73iT^{2} \) |
| 79 | \( 1 + (3.57 + 3.57i)T + 79iT^{2} \) |
| 83 | \( 1 + 3.58iT - 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 + (4.92 - 4.92i)T - 97iT^{2} \) |
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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
−15.17074458995310432444567510042, −13.71895947333878974007550448317, −13.28839778436496098482745670303, −11.36778488857008855211238513064, −10.16259663421182381543220163336, −8.559976702924126159326895352796, −8.149157095639185608129496376654, −6.23710216114107452558859306894, −5.59709554769783033229486120873, −4.40170198694495347286724613915,
1.36563827234338205544906405839, 3.25361316616112751865260897916, 4.89271477085151024316889919946, 6.67603997415276975541573465569, 8.484574285480603727748938226460, 9.811487014441597487893363119817, 10.78221570045610602864840604547, 11.51112950842836935700911668658, 12.51727732887984269458626764163, 13.37438224957663163590749953527