| L(s) = 1 | + 5-s − 7-s + 6·11-s + 13-s + 19-s − 6·23-s + 25-s + 6·29-s + 8·31-s − 35-s + 7·37-s + 6·41-s + 4·43-s − 12·47-s − 6·49-s − 6·53-s + 6·55-s − 11·61-s + 65-s + 7·67-s + 6·71-s + 11·73-s − 6·77-s − 79-s + 6·83-s + 12·89-s − 91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s + 1.80·11-s + 0.277·13-s + 0.229·19-s − 1.25·23-s + 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.169·35-s + 1.15·37-s + 0.937·41-s + 0.609·43-s − 1.75·47-s − 6/7·49-s − 0.824·53-s + 0.809·55-s − 1.40·61-s + 0.124·65-s + 0.855·67-s + 0.712·71-s + 1.28·73-s − 0.683·77-s − 0.112·79-s + 0.658·83-s + 1.27·89-s − 0.104·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.637496413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.637496413\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−7.940261201286850035959740619605, −6.82632434156042982882932493419, −6.28792086557716254895570026649, −6.08246200466970540807966057971, −4.87189282566873097543050531483, −4.24691358375935965316199398376, −3.51030194183122873712386088957, −2.66873289601469691257057755657, −1.65041632012686290686476457064, −0.843389572586074672489788769470,
0.843389572586074672489788769470, 1.65041632012686290686476457064, 2.66873289601469691257057755657, 3.51030194183122873712386088957, 4.24691358375935965316199398376, 4.87189282566873097543050531483, 6.08246200466970540807966057971, 6.28792086557716254895570026649, 6.82632434156042982882932493419, 7.940261201286850035959740619605
