| L(s) = 1 | + 3-s + 5-s + 9-s − 4·11-s + 6·13-s + 15-s − 6·17-s − 4·19-s + 25-s + 27-s − 2·29-s − 8·31-s − 4·33-s − 2·37-s + 6·39-s − 6·41-s + 12·43-s + 45-s + 8·47-s − 7·49-s − 6·51-s + 6·53-s − 4·55-s − 4·57-s + 12·59-s + 14·61-s + 6·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s + 1.66·13-s + 0.258·15-s − 1.45·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.696·33-s − 0.328·37-s + 0.960·39-s − 0.937·41-s + 1.82·43-s + 0.149·45-s + 1.16·47-s − 49-s − 0.840·51-s + 0.824·53-s − 0.539·55-s − 0.529·57-s + 1.56·59-s + 1.79·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.269494277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.269494277\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.23486091060546125236692720333, −13.00221841062232627578538988126, −11.15225413084276142148114637688, −10.45550448318589263597286780431, −9.042960162718976648409639393560, −8.323057177226078640880314002796, −6.89270810769313104844169224865, −5.59585158632682227968586342616, −3.95303834850677407956821293662, −2.25002432004047770108316173263,
2.25002432004047770108316173263, 3.95303834850677407956821293662, 5.59585158632682227968586342616, 6.89270810769313104844169224865, 8.323057177226078640880314002796, 9.042960162718976648409639393560, 10.45550448318589263597286780431, 11.15225413084276142148114637688, 13.00221841062232627578538988126, 13.23486091060546125236692720333
