| L(s) = 1 | + 3·3-s + 6·9-s − 3·13-s + 4·17-s + 4·19-s + 23-s + 9·27-s + 29-s − 31-s + 8·37-s − 9·39-s − 11·41-s + 10·43-s − 47-s + 12·51-s + 6·53-s + 12·57-s + 8·59-s + 8·61-s − 12·67-s + 3·69-s + 13·71-s + 7·73-s − 12·79-s + 9·81-s + 16·83-s + 3·87-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 2·9-s − 0.832·13-s + 0.970·17-s + 0.917·19-s + 0.208·23-s + 1.73·27-s + 0.185·29-s − 0.179·31-s + 1.31·37-s − 1.44·39-s − 1.71·41-s + 1.52·43-s − 0.145·47-s + 1.68·51-s + 0.824·53-s + 1.58·57-s + 1.04·59-s + 1.02·61-s − 1.46·67-s + 0.361·69-s + 1.54·71-s + 0.819·73-s − 1.35·79-s + 81-s + 1.75·83-s + 0.321·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.435086127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.435086127\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.65295084138705, −13.31067897518266, −12.79331041141711, −12.25957003817576, −11.83305689850557, −11.24242938591991, −10.43031949148581, −10.04909682935870, −9.620970436345945, −9.226865839648556, −8.725786838607868, −8.090985982244000, −7.810541169246212, −7.279435494080785, −6.921813225976429, −6.124933022315907, −5.366550988463609, −4.941942008422330, −4.196435223573727, −3.654045781038888, −3.208738847399882, −2.567313545466631, −2.212351558130903, −1.358233451891509, −0.7226855481872929,
0.7226855481872929, 1.358233451891509, 2.212351558130903, 2.567313545466631, 3.208738847399882, 3.654045781038888, 4.196435223573727, 4.941942008422330, 5.366550988463609, 6.124933022315907, 6.921813225976429, 7.279435494080785, 7.810541169246212, 8.090985982244000, 8.725786838607868, 9.226865839648556, 9.620970436345945, 10.04909682935870, 10.43031949148581, 11.24242938591991, 11.83305689850557, 12.25957003817576, 12.79331041141711, 13.31067897518266, 13.65295084138705
