L(s) = 1 | − 2·2-s + 2·4-s − 5-s − 7-s + 2·10-s + 2·11-s − 13-s + 2·14-s − 4·16-s + 19-s − 2·20-s − 4·22-s − 3·23-s − 4·25-s + 2·26-s − 2·28-s + 5·29-s + 9·31-s + 8·32-s + 35-s − 2·38-s − 2·41-s − 43-s + 4·44-s + 6·46-s − 3·47-s + 49-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s − 0.447·5-s − 0.377·7-s + 0.632·10-s + 0.603·11-s − 0.277·13-s + 0.534·14-s − 16-s + 0.229·19-s − 0.447·20-s − 0.852·22-s − 0.625·23-s − 4/5·25-s + 0.392·26-s − 0.377·28-s + 0.928·29-s + 1.61·31-s + 1.41·32-s + 0.169·35-s − 0.324·38-s − 0.312·41-s − 0.152·43-s + 0.603·44-s + 0.884·46-s − 0.437·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5987677264\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5987677264\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 17 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.930199833374644671352816709179, −9.532660112777972535438596537729, −8.462043055034753096507851091821, −7.985150749865692086998616861589, −6.99822189829147465145755873932, −6.27531093268500063877449025336, −4.80346894431881084860452543382, −3.70219368766936415572720638719, −2.25590170395299068912333781526, −0.77824920253834103627110112541,
0.77824920253834103627110112541, 2.25590170395299068912333781526, 3.70219368766936415572720638719, 4.80346894431881084860452543382, 6.27531093268500063877449025336, 6.99822189829147465145755873932, 7.985150749865692086998616861589, 8.462043055034753096507851091821, 9.532660112777972535438596537729, 9.930199833374644671352816709179