| L(s) = 1 | − 3-s − 2·5-s + 9-s + 4·11-s + 2·13-s + 2·15-s + 6·17-s + 4·19-s − 25-s − 27-s + 2·29-s − 4·31-s − 4·33-s − 2·37-s − 2·39-s − 2·41-s + 4·43-s − 2·45-s − 8·47-s − 6·51-s + 10·53-s − 8·55-s − 4·57-s + 4·59-s − 6·61-s − 4·65-s + 4·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.516·15-s + 1.45·17-s + 0.917·19-s − 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.696·33-s − 0.328·37-s − 0.320·39-s − 0.312·41-s + 0.609·43-s − 0.298·45-s − 1.16·47-s − 0.840·51-s + 1.37·53-s − 1.07·55-s − 0.529·57-s + 0.520·59-s − 0.768·61-s − 0.496·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.513486738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.513486738\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−8.222151966740788314456121575198, −7.47837202269369036994254596914, −6.96458755801090666740322092908, −6.03898188878233827131141159382, −5.47373462538208654239189473772, −4.51006367899790870186381301422, −3.73528919773956898681806633984, −3.22568568252812840477382664374, −1.61711448111568648295505994452, −0.75476969164145883416664038839,
0.75476969164145883416664038839, 1.61711448111568648295505994452, 3.22568568252812840477382664374, 3.73528919773956898681806633984, 4.51006367899790870186381301422, 5.47373462538208654239189473772, 6.03898188878233827131141159382, 6.96458755801090666740322092908, 7.47837202269369036994254596914, 8.222151966740788314456121575198
