L(s) = 1 | − 2-s + 5-s − 1.41i·7-s + 8-s − 10-s + 11-s + 1.41i·14-s − 16-s − 1.41i·19-s − 22-s + 1.41i·29-s − 31-s − 1.41i·35-s + 37-s + 1.41i·38-s + ⋯ |
L(s) = 1 | − 2-s + 5-s − 1.41i·7-s + 8-s − 10-s + 11-s + 1.41i·14-s − 16-s − 1.41i·19-s − 22-s + 1.41i·29-s − 31-s − 1.41i·35-s + 37-s + 1.41i·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1557 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1557 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7522793412\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7522793412\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 173 | \( 1 - T \) |
good | 2 | \( 1 + T + T^{2} \) |
| 5 | \( 1 - T + T^{2} \) |
| 7 | \( 1 + 1.41iT - T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + 1.41iT - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 1.41iT - T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + 1.41iT - T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 - 1.41iT - T^{2} \) |
| 83 | \( 1 - 1.41iT - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + 1.41iT - T^{2} \) |
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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
−9.565835919330122940495317411516, −8.918049519468224049440511336496, −8.062079165240402230773928756239, −6.95829102912759421221076918846, −6.79875023021112684648148062836, −5.36843104611759082607566228157, −4.48356734584018720913082804307, −3.55341216909524798486923754681, −1.93390413920784301225517886575, −0.947627843752204265299253020835,
1.54176698445586549480915237533, 2.27535401652704931647262932643, 3.76128663308943664194000900046, 4.93519528013636861941628711067, 5.92069177501044245728103029382, 6.34444248279772160107631369976, 7.70225793479821420465170857462, 8.324959752860125911620926140612, 9.230048199855122154226755921513, 9.490355893158025642147342753944