| L(s) = 1 | + 2·3-s + 2·5-s + 7-s + 9-s − 4·13-s + 4·15-s − 6·19-s + 2·21-s + 3·23-s − 25-s − 4·27-s − 5·29-s − 2·31-s + 2·35-s + 3·37-s − 8·39-s + 7·43-s + 2·45-s + 49-s − 53-s − 12·57-s − 14·59-s − 2·61-s + 63-s − 8·65-s + 4·67-s + 6·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.10·13-s + 1.03·15-s − 1.37·19-s + 0.436·21-s + 0.625·23-s − 1/5·25-s − 0.769·27-s − 0.928·29-s − 0.359·31-s + 0.338·35-s + 0.493·37-s − 1.28·39-s + 1.06·43-s + 0.298·45-s + 1/7·49-s − 0.137·53-s − 1.58·57-s − 1.82·59-s − 0.256·61-s + 0.125·63-s − 0.992·65-s + 0.488·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 11 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.25067082984995, −15.30887864968814, −15.06341209054441, −14.42089355793226, −14.22799572756187, −13.50177160384999, −13.00441779552708, −12.58500000924708, −11.77881390917316, −11.07278274288795, −10.51835544632158, −9.839982563692878, −9.304126721785830, −8.935198351344527, −8.289761583117375, −7.555829306176327, −7.243072051715566, −6.187647672101114, −5.777959124239697, −4.887100829797342, −4.303584199712500, −3.458863328831882, −2.609836530048954, −2.198446929618027, −1.511630529097296, 0,
1.511630529097296, 2.198446929618027, 2.609836530048954, 3.458863328831882, 4.303584199712500, 4.887100829797342, 5.777959124239697, 6.187647672101114, 7.243072051715566, 7.555829306176327, 8.289761583117375, 8.935198351344527, 9.304126721785830, 9.839982563692878, 10.51835544632158, 11.07278274288795, 11.77881390917316, 12.58500000924708, 13.00441779552708, 13.50177160384999, 14.22799572756187, 14.42089355793226, 15.06341209054441, 15.30887864968814, 16.25067082984995
