| L(s) = 1 | − 2·3-s + 4·5-s + 7-s + 9-s − 8·15-s − 2·17-s + 2·19-s − 2·21-s + 8·23-s + 11·25-s + 4·27-s − 2·29-s + 4·31-s + 4·35-s + 6·37-s − 2·41-s − 8·43-s + 4·45-s − 4·47-s + 49-s + 4·51-s + 10·53-s − 4·57-s − 6·59-s − 4·61-s + 63-s + 12·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s + 0.377·7-s + 1/3·9-s − 2.06·15-s − 0.485·17-s + 0.458·19-s − 0.436·21-s + 1.66·23-s + 11/5·25-s + 0.769·27-s − 0.371·29-s + 0.718·31-s + 0.676·35-s + 0.986·37-s − 0.312·41-s − 1.21·43-s + 0.596·45-s − 0.583·47-s + 1/7·49-s + 0.560·51-s + 1.37·53-s − 0.529·57-s − 0.781·59-s − 0.512·61-s + 0.125·63-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.340701490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.340701490\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 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
−11.04646830243163367622011090887, −10.30690858233962784710059471081, −9.483344466014645468218899692437, −8.589748372800265236248597894315, −7.00881059716814921000381829301, −6.25148198523199051609095248236, −5.43110360120582172209460233834, −4.80018849900442151332658226553, −2.73137510590455017802711867793, −1.29582036582827735540991947956,
1.29582036582827735540991947956, 2.73137510590455017802711867793, 4.80018849900442151332658226553, 5.43110360120582172209460233834, 6.25148198523199051609095248236, 7.00881059716814921000381829301, 8.589748372800265236248597894315, 9.483344466014645468218899692437, 10.30690858233962784710059471081, 11.04646830243163367622011090887
