| L(s) = 1 | + 2·3-s + 9-s + 11-s + 2·13-s + 4·17-s − 5·23-s − 4·27-s + 3·29-s + 10·31-s + 2·33-s + 5·37-s + 4·39-s + 10·41-s − 5·43-s − 4·47-s + 8·51-s + 10·53-s − 10·59-s + 10·61-s + 5·67-s − 10·69-s + 3·71-s − 10·73-s + 13·79-s − 11·81-s + 10·83-s + 6·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 0.301·11-s + 0.554·13-s + 0.970·17-s − 1.04·23-s − 0.769·27-s + 0.557·29-s + 1.79·31-s + 0.348·33-s + 0.821·37-s + 0.640·39-s + 1.56·41-s − 0.762·43-s − 0.583·47-s + 1.12·51-s + 1.37·53-s − 1.30·59-s + 1.28·61-s + 0.610·67-s − 1.20·69-s + 0.356·71-s − 1.17·73-s + 1.46·79-s − 1.22·81-s + 1.09·83-s + 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.659336591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.659336591\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.01060201494176, −13.64350964572689, −13.22577651977870, −12.55485662971844, −12.01678298471088, −11.64442962375139, −11.03061142097297, −10.35664187581374, −9.846466174885191, −9.546060990509069, −8.862164510424405, −8.402641573456657, −7.947537811455641, −7.677931457220838, −6.830530154329250, −6.254792588565977, −5.828581364436808, −5.099942374422420, −4.320133312522161, −3.893723441126856, −3.283394612282300, −2.710850220093964, −2.198880124522779, −1.348177094090132, −0.6861897958478683,
0.6861897958478683, 1.348177094090132, 2.198880124522779, 2.710850220093964, 3.283394612282300, 3.893723441126856, 4.320133312522161, 5.099942374422420, 5.828581364436808, 6.254792588565977, 6.830530154329250, 7.677931457220838, 7.947537811455641, 8.402641573456657, 8.862164510424405, 9.546060990509069, 9.846466174885191, 10.35664187581374, 11.03061142097297, 11.64442962375139, 12.01678298471088, 12.55485662971844, 13.22577651977870, 13.64350964572689, 14.01060201494176
