| L(s) = 1 | + 3-s + 2·7-s + 9-s + 13-s − 6·17-s + 2·19-s + 2·21-s − 5·25-s + 27-s − 6·29-s + 2·31-s + 2·37-s + 39-s − 12·41-s − 4·43-s − 3·49-s − 6·51-s + 6·53-s + 2·57-s + 12·59-s + 2·61-s + 2·63-s − 10·67-s + 12·71-s + 14·73-s − 5·75-s + 8·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.277·13-s − 1.45·17-s + 0.458·19-s + 0.436·21-s − 25-s + 0.192·27-s − 1.11·29-s + 0.359·31-s + 0.328·37-s + 0.160·39-s − 1.87·41-s − 0.609·43-s − 3/7·49-s − 0.840·51-s + 0.824·53-s + 0.264·57-s + 1.56·59-s + 0.256·61-s + 0.251·63-s − 1.22·67-s + 1.42·71-s + 1.63·73-s − 0.577·75-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.376149593\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.376149593\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.21941002116462755470673769512, −11.83258307695703192104822918116, −11.02368824085056898890395801496, −9.798663464890123710839719924879, −8.735814680045936069504206621070, −7.86871640882441783507560416572, −6.65843891177652373785895186115, −5.11454570698520102977754794044, −3.80067854107620357152276821608, −2.03881758362108754018941589683,
2.03881758362108754018941589683, 3.80067854107620357152276821608, 5.11454570698520102977754794044, 6.65843891177652373785895186115, 7.86871640882441783507560416572, 8.735814680045936069504206621070, 9.798663464890123710839719924879, 11.02368824085056898890395801496, 11.83258307695703192104822918116, 13.21941002116462755470673769512
