| L(s) = 1 | + 2.56·2-s + 3-s + 4.56·4-s − 5-s + 2.56·6-s − 7-s + 6.56·8-s + 9-s − 2.56·10-s + 3.12·11-s + 4.56·12-s + 3.56·13-s − 2.56·14-s − 15-s + 7.68·16-s − 17-s + 2.56·18-s − 7.12·19-s − 4.56·20-s − 21-s + 8·22-s + 8.68·23-s + 6.56·24-s + 25-s + 9.12·26-s + 27-s − 4.56·28-s + ⋯ |
| L(s) = 1 | + 1.81·2-s + 0.577·3-s + 2.28·4-s − 0.447·5-s + 1.04·6-s − 0.377·7-s + 2.31·8-s + 0.333·9-s − 0.810·10-s + 0.941·11-s + 1.31·12-s + 0.987·13-s − 0.684·14-s − 0.258·15-s + 1.92·16-s − 0.242·17-s + 0.603·18-s − 1.63·19-s − 1.01·20-s − 0.218·21-s + 1.70·22-s + 1.81·23-s + 1.33·24-s + 0.200·25-s + 1.78·26-s + 0.192·27-s − 0.862·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.178406453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.178406453\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - 2.56T + 2T^{2} \) |
| 11 | \( 1 - 3.12T + 11T^{2} \) |
| 13 | \( 1 - 3.56T + 13T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 - 8.68T + 23T^{2} \) |
| 29 | \( 1 + 1.12T + 29T^{2} \) |
| 31 | \( 1 - 2.43T + 31T^{2} \) |
| 37 | \( 1 + 3.56T + 37T^{2} \) |
| 41 | \( 1 - 3.56T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 + 2.87T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 + 6.24T + 67T^{2} \) |
| 71 | \( 1 + 7.12T + 71T^{2} \) |
| 73 | \( 1 - 2.87T + 73T^{2} \) |
| 79 | \( 1 - 3.12T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
| 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
−9.080658977392302749807939672320, −8.499489019713316489828287440884, −7.33749952364122450869751170006, −6.59294025916800309965365867661, −6.16043648869638976910396516887, −4.92432828712169903519685640178, −4.19363900142589069621769757381, −3.54488608103195909899912214561, −2.79062541351769277558221454342, −1.55971063092187508187260055572,
1.55971063092187508187260055572, 2.79062541351769277558221454342, 3.54488608103195909899912214561, 4.19363900142589069621769757381, 4.92432828712169903519685640178, 6.16043648869638976910396516887, 6.59294025916800309965365867661, 7.33749952364122450869751170006, 8.499489019713316489828287440884, 9.080658977392302749807939672320
