L(s) = 1 | + 0.692i·2-s + (−1.66 + 0.465i)3-s + 1.51·4-s + (0.5 − 0.866i)5-s + (−0.322 − 1.15i)6-s + (−0.669 − 2.55i)7-s + 2.43i·8-s + (2.56 − 1.55i)9-s + (0.600 + 0.346i)10-s + (1.74 − 1.00i)11-s + (−2.53 + 0.707i)12-s + (3.66 − 2.11i)13-s + (1.77 − 0.463i)14-s + (−0.431 + 1.67i)15-s + 1.35·16-s + (−1.55 + 2.69i)17-s + ⋯ |
L(s) = 1 | + 0.489i·2-s + (−0.963 + 0.268i)3-s + 0.759·4-s + (0.223 − 0.387i)5-s + (−0.131 − 0.471i)6-s + (−0.253 − 0.967i)7-s + 0.862i·8-s + (0.855 − 0.517i)9-s + (0.189 + 0.109i)10-s + (0.527 − 0.304i)11-s + (−0.732 + 0.204i)12-s + (1.01 − 0.586i)13-s + (0.473 − 0.123i)14-s + (−0.111 + 0.433i)15-s + 0.337·16-s + (−0.376 + 0.652i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.226i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.26202 + 0.144800i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.26202 + 0.144800i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.66 - 0.465i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.669 + 2.55i)T \) |
good | 2 | \( 1 - 0.692iT - 2T^{2} \) |
| 11 | \( 1 + (-1.74 + 1.00i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.66 + 2.11i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.55 - 2.69i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.112 - 0.0652i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.326 - 0.188i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-7.55 - 4.35i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 3.61iT - 31T^{2} \) |
| 37 | \( 1 + (-2.94 - 5.09i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.02 + 1.77i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.79 + 8.30i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 4.62T + 47T^{2} \) |
| 53 | \( 1 + (8.28 + 4.78i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 5.36iT - 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 7.48iT - 71T^{2} \) |
| 73 | \( 1 + (1.34 + 0.777i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 2.71T + 79T^{2} \) |
| 83 | \( 1 + (0.308 - 0.534i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.725 - 1.25i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-11.3 - 6.55i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.52272537908195903367784213030, −10.76592852020514235842533712359, −10.17052956422588008320401328470, −8.785027676429155992501833883797, −7.66237790373752945816416292407, −6.49622222437412429424366688793, −6.10337815193223465793058684719, −4.81678134509684196279024213463, −3.53683201448118237740778868684, −1.25351789627817076170672159764,
1.57510437801153906653426735042, 2.88389406861596535760049958606, 4.52471467389957708573989536974, 6.16091498165501051834604236813, 6.36272867870957042941743119152, 7.52924945890981752248267775733, 9.060485199711449716164137014017, 10.01420080675887575773344625611, 10.99583335356287447692009121313, 11.58347378574227894440494729841