| L(s) = 1 | + 3·3-s − 5-s − 7-s + 6·9-s − 5·11-s − 3·13-s − 3·15-s − 17-s + 6·19-s − 3·21-s + 6·23-s + 25-s + 9·27-s − 9·29-s − 4·31-s − 15·33-s + 35-s + 2·37-s − 9·39-s − 4·41-s + 10·43-s − 6·45-s − 47-s + 49-s − 3·51-s + 4·53-s + 5·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s − 0.377·7-s + 2·9-s − 1.50·11-s − 0.832·13-s − 0.774·15-s − 0.242·17-s + 1.37·19-s − 0.654·21-s + 1.25·23-s + 1/5·25-s + 1.73·27-s − 1.67·29-s − 0.718·31-s − 2.61·33-s + 0.169·35-s + 0.328·37-s − 1.44·39-s − 0.624·41-s + 1.52·43-s − 0.894·45-s − 0.145·47-s + 1/7·49-s − 0.420·51-s + 0.549·53-s + 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.555511632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.555511632\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
| 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
−13.22652980823546667505573341590, −12.58088516701521350060452238667, −10.97393679911058735992196935543, −9.743659591392653928139921572811, −9.025895505037624522078550270792, −7.73731776455161087713955528993, −7.33986466138726521878856236101, −5.11248277682454093810005074849, −3.51475005035302395747602864778, −2.51163217316890307126702952381,
2.51163217316890307126702952381, 3.51475005035302395747602864778, 5.11248277682454093810005074849, 7.33986466138726521878856236101, 7.73731776455161087713955528993, 9.025895505037624522078550270792, 9.743659591392653928139921572811, 10.97393679911058735992196935543, 12.58088516701521350060452238667, 13.22652980823546667505573341590
