| L(s) = 1 | + 3·7-s − 2·11-s + 2·13-s + 5·17-s − 2·19-s − 8·23-s − 10·37-s − 3·41-s + 2·43-s − 9·47-s + 2·49-s + 12·53-s − 6·59-s + 2·61-s + 12·67-s + 12·71-s − 13·73-s − 6·77-s − 79-s − 4·83-s + 7·89-s + 6·91-s + 97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.13·7-s − 0.603·11-s + 0.554·13-s + 1.21·17-s − 0.458·19-s − 1.66·23-s − 1.64·37-s − 0.468·41-s + 0.304·43-s − 1.31·47-s + 2/7·49-s + 1.64·53-s − 0.781·59-s + 0.256·61-s + 1.46·67-s + 1.42·71-s − 1.52·73-s − 0.683·77-s − 0.112·79-s − 0.439·83-s + 0.741·89-s + 0.628·91-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−14.35720906410047, −14.10270259674807, −13.56649312753488, −12.96284857279840, −12.43131381342055, −11.78366371023441, −11.66994266956723, −10.82159539912518, −10.48274649942577, −10.01036065297914, −9.448268602194141, −8.571946010162535, −8.292433552278649, −7.922272695374461, −7.304417210538045, −6.689892894125800, −5.949754136276323, −5.494452523000389, −5.008739455313877, −4.356646265266708, −3.701039579226852, −3.197435884620142, −2.184761312507507, −1.801751477178448, −1.013948729070745, 0,
1.013948729070745, 1.801751477178448, 2.184761312507507, 3.197435884620142, 3.701039579226852, 4.356646265266708, 5.008739455313877, 5.494452523000389, 5.949754136276323, 6.689892894125800, 7.304417210538045, 7.922272695374461, 8.292433552278649, 8.571946010162535, 9.448268602194141, 10.01036065297914, 10.48274649942577, 10.82159539912518, 11.66994266956723, 11.78366371023441, 12.43131381342055, 12.96284857279840, 13.56649312753488, 14.10270259674807, 14.35720906410047
