| L(s) = 1 | + 3-s − 2·7-s + 9-s − 2·11-s − 2·13-s + 4·17-s + 2·19-s − 2·21-s − 2·23-s + 27-s + 6·29-s + 8·31-s − 2·33-s + 10·37-s − 2·39-s + 2·41-s + 43-s + 6·47-s − 3·49-s + 4·51-s − 6·53-s + 2·57-s − 6·59-s − 2·63-s + 12·67-s − 2·69-s − 8·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s + 0.970·17-s + 0.458·19-s − 0.436·21-s − 0.417·23-s + 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.348·33-s + 1.64·37-s − 0.320·39-s + 0.312·41-s + 0.152·43-s + 0.875·47-s − 3/7·49-s + 0.560·51-s − 0.824·53-s + 0.264·57-s − 0.781·59-s − 0.251·63-s + 1.46·67-s − 0.240·69-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.387961301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.387961301\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−12.93077750252840, −12.70160875299195, −12.12350012918252, −11.78606649125165, −11.18116173278387, −10.45727108868172, −10.14307911905241, −9.790104791379446, −9.318546332521682, −8.870258013999370, −8.114969172001615, −7.725857414129052, −7.630079715141762, −6.667129543915444, −6.385101054595460, −5.859232080923797, −5.140544392217033, −4.731037703461070, −4.133787981878064, −3.489000827869070, −2.917182332043511, −2.655672993687905, −1.976394105517736, −1.022804545680732, −0.5740534954591430,
0.5740534954591430, 1.022804545680732, 1.976394105517736, 2.655672993687905, 2.917182332043511, 3.489000827869070, 4.133787981878064, 4.731037703461070, 5.140544392217033, 5.859232080923797, 6.385101054595460, 6.667129543915444, 7.630079715141762, 7.725857414129052, 8.114969172001615, 8.870258013999370, 9.318546332521682, 9.790104791379446, 10.14307911905241, 10.45727108868172, 11.18116173278387, 11.78606649125165, 12.12350012918252, 12.70160875299195, 12.93077750252840
