| L(s) = 1 | − 2·7-s − 6·11-s + 13-s − 6·17-s − 6·19-s − 5·23-s − 7·29-s + 4·31-s + 10·37-s + 2·41-s − 3·43-s + 2·47-s − 3·49-s − 9·53-s + 10·59-s − 7·61-s + 2·67-s − 12·71-s + 6·73-s + 12·77-s + 9·79-s + 2·83-s − 4·89-s − 2·91-s − 14·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1.80·11-s + 0.277·13-s − 1.45·17-s − 1.37·19-s − 1.04·23-s − 1.29·29-s + 0.718·31-s + 1.64·37-s + 0.312·41-s − 0.457·43-s + 0.291·47-s − 3/7·49-s − 1.23·53-s + 1.30·59-s − 0.896·61-s + 0.244·67-s − 1.42·71-s + 0.702·73-s + 1.36·77-s + 1.01·79-s + 0.219·83-s − 0.423·89-s − 0.209·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.24203785206699, −13.01013993820852, −12.61595878559306, −12.02817636891431, −11.31154162417900, −10.92362684485340, −10.67115860988466, −10.01072865436677, −9.647525692525284, −9.158896484014951, −8.452367317199698, −8.159831033137872, −7.713310526384537, −7.046965077286754, −6.562145522224525, −5.949632125965775, −5.826069069169063, −4.885294132990204, −4.519250030287770, −3.981768280645184, −3.325588959034488, −2.605565713975658, −2.317857547362829, −1.700874458268676, −0.4933338352800913, 0,
0.4933338352800913, 1.700874458268676, 2.317857547362829, 2.605565713975658, 3.325588959034488, 3.981768280645184, 4.519250030287770, 4.885294132990204, 5.826069069169063, 5.949632125965775, 6.562145522224525, 7.046965077286754, 7.713310526384537, 8.159831033137872, 8.452367317199698, 9.158896484014951, 9.647525692525284, 10.01072865436677, 10.67115860988466, 10.92362684485340, 11.31154162417900, 12.02817636891431, 12.61595878559306, 13.01013993820852, 13.24203785206699
