| L(s) = 1 | + 2·2-s − 3-s + 2·4-s + 5-s − 2·6-s − 7-s + 9-s + 2·10-s − 11-s − 2·12-s − 3·13-s − 2·14-s − 15-s − 4·16-s + 2·18-s + 4·19-s + 2·20-s + 21-s − 2·22-s + 3·23-s + 25-s − 6·26-s − 27-s − 2·28-s + 2·29-s − 2·30-s − 3·31-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s − 0.377·7-s + 1/3·9-s + 0.632·10-s − 0.301·11-s − 0.577·12-s − 0.832·13-s − 0.534·14-s − 0.258·15-s − 16-s + 0.471·18-s + 0.917·19-s + 0.447·20-s + 0.218·21-s − 0.426·22-s + 0.625·23-s + 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.377·28-s + 0.371·29-s − 0.365·30-s − 0.538·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 333795 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333795 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.495412795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.495412795\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−12.56389799935493, −12.35602439804562, −11.73811537543228, −11.52133471112349, −10.81763438649098, −10.52131590393102, −9.846013775885578, −9.579836695306973, −8.965620470243096, −8.590930177497837, −7.640207394976733, −7.326571578094757, −6.862785583309017, −6.397033093184449, −5.796042959859885, −5.523311556939932, −5.044050977941784, −4.717513304489239, −4.005975923458593, −3.654943209573034, −2.819899782117303, −2.679410491043232, −1.955167741676410, −1.144727726759823, −0.4069178322067818,
0.4069178322067818, 1.144727726759823, 1.955167741676410, 2.679410491043232, 2.819899782117303, 3.654943209573034, 4.005975923458593, 4.717513304489239, 5.044050977941784, 5.523311556939932, 5.796042959859885, 6.397033093184449, 6.862785583309017, 7.326571578094757, 7.640207394976733, 8.590930177497837, 8.965620470243096, 9.579836695306973, 9.846013775885578, 10.52131590393102, 10.81763438649098, 11.52133471112349, 11.73811537543228, 12.35602439804562, 12.56389799935493
