L(s) = 1 | − 1.26i·2-s + i·3-s + 0.399·4-s + i·5-s + 1.26·6-s + (−0.0556 − 2.64i)7-s − 3.03i·8-s − 9-s + 1.26·10-s + (−0.516 − 3.27i)11-s + 0.399i·12-s − 6.21·13-s + (−3.34 + 0.0703i)14-s − 15-s − 3.04·16-s − 1.62·17-s + ⋯ |
L(s) = 1 | − 0.894i·2-s + 0.577i·3-s + 0.199·4-s + 0.447i·5-s + 0.516·6-s + (−0.0210 − 0.999i)7-s − 1.07i·8-s − 0.333·9-s + 0.400·10-s + (−0.155 − 0.987i)11-s + 0.115i·12-s − 1.72·13-s + (−0.894 + 0.0188i)14-s − 0.258·15-s − 0.760·16-s − 0.394·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.176i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9180170632\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9180170632\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (0.0556 + 2.64i)T \) |
| 11 | \( 1 + (0.516 + 3.27i)T \) |
good | 2 | \( 1 + 1.26iT - 2T^{2} \) |
| 13 | \( 1 + 6.21T + 13T^{2} \) |
| 17 | \( 1 + 1.62T + 17T^{2} \) |
| 19 | \( 1 + 4.38T + 19T^{2} \) |
| 23 | \( 1 + 1.24T + 23T^{2} \) |
| 29 | \( 1 - 9.91iT - 29T^{2} \) |
| 31 | \( 1 + 7.78iT - 31T^{2} \) |
| 37 | \( 1 - 1.61T + 37T^{2} \) |
| 41 | \( 1 - 4.95T + 41T^{2} \) |
| 43 | \( 1 + 3.35iT - 43T^{2} \) |
| 47 | \( 1 + 4.74iT - 47T^{2} \) |
| 53 | \( 1 + 8.51T + 53T^{2} \) |
| 59 | \( 1 - 4.49iT - 59T^{2} \) |
| 61 | \( 1 - 6.43T + 61T^{2} \) |
| 67 | \( 1 - 3.96T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 + 8.16T + 73T^{2} \) |
| 79 | \( 1 + 13.4iT - 79T^{2} \) |
| 83 | \( 1 - 2.78T + 83T^{2} \) |
| 89 | \( 1 - 2.92iT - 89T^{2} \) |
| 97 | \( 1 + 15.7iT - 97T^{2} \) |
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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
−9.841622795533875782021726257072, −8.846666552420074172730670364819, −7.65813476365512892500556255238, −6.98132200978374220066870356389, −6.07474118034903881075544305400, −4.78538504407213385991267811303, −3.90557472856821504910825232370, −3.05684232264936008261732432914, −2.11382030136272731361383726755, −0.34979714002141833801342343303,
2.07541577303936242743016012276, 2.53066812387758159948083743397, 4.56716902364069653543320989894, 5.20369907262918504214766824501, 6.18047046990194290971786474010, 6.81654248849165644693015524701, 7.73130280901320768007722618544, 8.206015466107449140557494795178, 9.205470133885255613600115279728, 9.921949553805496543040599331178