L(s) = 1 | + 3-s − 7-s + 9-s + 2·11-s + 2·13-s + 4·17-s + 4·19-s − 21-s − 6·23-s − 5·25-s + 27-s + 2·29-s + 2·33-s + 6·37-s + 2·39-s + 8·41-s + 8·43-s − 4·47-s + 49-s + 4·51-s + 6·53-s + 4·57-s + 14·61-s − 63-s − 4·67-s − 6·69-s − 2·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s + 0.970·17-s + 0.917·19-s − 0.218·21-s − 1.25·23-s − 25-s + 0.192·27-s + 0.371·29-s + 0.348·33-s + 0.986·37-s + 0.320·39-s + 1.24·41-s + 1.21·43-s − 0.583·47-s + 1/7·49-s + 0.560·51-s + 0.824·53-s + 0.529·57-s + 1.79·61-s − 0.125·63-s − 0.488·67-s − 0.722·69-s − 0.237·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.160376678\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.160376678\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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
−9.694432620383059295044846371230, −8.839846973587512271832482159777, −7.955699455902827737318339326897, −7.36235129594086681027946055340, −6.24035827483151455450786911728, −5.60729594063914370947972765374, −4.20599887184774343980959477769, −3.57055610765599388652202709940, −2.47132763276648395687043734661, −1.12641799956446265250447029021,
1.12641799956446265250447029021, 2.47132763276648395687043734661, 3.57055610765599388652202709940, 4.20599887184774343980959477769, 5.60729594063914370947972765374, 6.24035827483151455450786911728, 7.36235129594086681027946055340, 7.955699455902827737318339326897, 8.839846973587512271832482159777, 9.694432620383059295044846371230