| L(s) = 1 | + 2-s − 7·4-s + 20·5-s − 2·7-s − 15·8-s + 20·10-s + 48·11-s − 14·13-s − 2·14-s + 41·16-s + 17·17-s + 92·19-s − 140·20-s + 48·22-s + 122·23-s + 275·25-s − 14·26-s + 14·28-s + 36·29-s − 182·31-s + 161·32-s + 17·34-s − 40·35-s + 76·37-s + 92·38-s − 300·40-s − 294·41-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 7/8·4-s + 1.78·5-s − 0.107·7-s − 0.662·8-s + 0.632·10-s + 1.31·11-s − 0.298·13-s − 0.0381·14-s + 0.640·16-s + 0.242·17-s + 1.11·19-s − 1.56·20-s + 0.465·22-s + 1.10·23-s + 11/5·25-s − 0.105·26-s + 0.0944·28-s + 0.230·29-s − 1.05·31-s + 0.889·32-s + 0.0857·34-s − 0.193·35-s + 0.337·37-s + 0.392·38-s − 1.18·40-s − 1.11·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.261351985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.261351985\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 - p T \) |
| good | 2 | \( 1 - T + p^{3} T^{2} \) |
| 5 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 7 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 - 48 T + p^{3} T^{2} \) |
| 13 | \( 1 + 14 T + p^{3} T^{2} \) |
| 19 | \( 1 - 92 T + p^{3} T^{2} \) |
| 23 | \( 1 - 122 T + p^{3} T^{2} \) |
| 29 | \( 1 - 36 T + p^{3} T^{2} \) |
| 31 | \( 1 + 182 T + p^{3} T^{2} \) |
| 37 | \( 1 - 76 T + p^{3} T^{2} \) |
| 41 | \( 1 + 294 T + p^{3} T^{2} \) |
| 43 | \( 1 + 428 T + p^{3} T^{2} \) |
| 47 | \( 1 - 12 T + p^{3} T^{2} \) |
| 53 | \( 1 - 234 T + p^{3} T^{2} \) |
| 59 | \( 1 - 540 T + p^{3} T^{2} \) |
| 61 | \( 1 + 820 T + p^{3} T^{2} \) |
| 67 | \( 1 - 700 T + p^{3} T^{2} \) |
| 71 | \( 1 + 794 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1038 T + p^{3} T^{2} \) |
| 79 | \( 1 - 858 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1052 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1102 T + p^{3} T^{2} \) |
| 97 | \( 1 - 710 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.84572032586095150129070280011, −11.68873954698695547292313537973, −10.12340851965310588877862971314, −9.468567662220988120960864940459, −8.777328738222094048189913970631, −6.87966353094760050551239872917, −5.78427121583277909087922347724, −4.90786977862438319421197028568, −3.24550987121121250197287421243, −1.38427781376150785635910606005,
1.38427781376150785635910606005, 3.24550987121121250197287421243, 4.90786977862438319421197028568, 5.78427121583277909087922347724, 6.87966353094760050551239872917, 8.777328738222094048189913970631, 9.468567662220988120960864940459, 10.12340851965310588877862971314, 11.68873954698695547292313537973, 12.84572032586095150129070280011
