| L(s) = 1 | − 1.76i·3-s + 8.20·5-s + 5.87·9-s + 10.5i·11-s + 5.87·13-s − 14.5i·15-s − 27.0·17-s − 12.3i·19-s − 41.9i·23-s + 42.2·25-s − 26.3i·27-s + 36.4·29-s + 7.64i·31-s + 18.5·33-s + 7.25·37-s + ⋯ |
| L(s) = 1 | − 0.589i·3-s + 1.64·5-s + 0.652·9-s + 0.955i·11-s + 0.451·13-s − 0.967i·15-s − 1.58·17-s − 0.647i·19-s − 1.82i·23-s + 1.69·25-s − 0.974i·27-s + 1.25·29-s + 0.246i·31-s + 0.563·33-s + 0.196·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.045578468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.045578468\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 1.76iT - 9T^{2} \) |
| 5 | \( 1 - 8.20T + 25T^{2} \) |
| 11 | \( 1 - 10.5iT - 121T^{2} \) |
| 13 | \( 1 - 5.87T + 169T^{2} \) |
| 17 | \( 1 + 27.0T + 289T^{2} \) |
| 19 | \( 1 + 12.3iT - 361T^{2} \) |
| 23 | \( 1 + 41.9iT - 529T^{2} \) |
| 29 | \( 1 - 36.4T + 841T^{2} \) |
| 31 | \( 1 - 7.64iT - 961T^{2} \) |
| 37 | \( 1 - 7.25T + 1.36e3T^{2} \) |
| 41 | \( 1 - 48.1T + 1.68e3T^{2} \) |
| 43 | \( 1 - 23.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 59.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 4.06T + 2.80e3T^{2} \) |
| 59 | \( 1 - 26.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 0.370T + 3.72e3T^{2} \) |
| 67 | \( 1 + 2.08iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 121. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 77.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 36.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 98.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 30.3T + 7.92e3T^{2} \) |
| 97 | \( 1 - 5.87T + 9.40e3T^{2} \) |
| 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
−9.094285813998507357236575002265, −8.558597662253864207816641051285, −7.29282727677850763009994288039, −6.53287319144029525219152994477, −6.27156110662465781266578525644, −4.92235331727106541157958683393, −4.35705263875738290081507153844, −2.51090266755196304743512274390, −2.08782257801348456766684296828, −0.929372061749576014449283424632,
1.19510657100474108319101604816, 2.16937215793739853827687520722, 3.33731053959229610234761154586, 4.36818163070755102673846689061, 5.31157554167878777362921309554, 6.06084479943003344983399056721, 6.64405246450234180495243958290, 7.82096864853338927159491098879, 8.906374640329482227098231366409, 9.368358792333459014871381781358
