L(s) = 1 | + (0.173 − 0.984i)4-s + (−0.5 + 0.866i)7-s + (−0.592 − 1.62i)13-s + (−0.939 − 0.342i)16-s + (0.939 − 0.342i)25-s + (0.766 + 0.642i)28-s + (−1.5 − 0.866i)31-s − 1.73i·37-s + (−0.173 − 0.984i)43-s + (−1.70 + 0.300i)52-s + (0.173 − 0.984i)61-s + (−0.5 + 0.866i)64-s + (1.11 − 1.32i)67-s + (0.939 + 0.342i)73-s + (−0.592 + 1.62i)79-s + ⋯ |
L(s) = 1 | + (0.173 − 0.984i)4-s + (−0.5 + 0.866i)7-s + (−0.592 − 1.62i)13-s + (−0.939 − 0.342i)16-s + (0.939 − 0.342i)25-s + (0.766 + 0.642i)28-s + (−1.5 − 0.866i)31-s − 1.73i·37-s + (−0.173 − 0.984i)43-s + (−1.70 + 0.300i)52-s + (0.173 − 0.984i)61-s + (−0.5 + 0.866i)64-s + (1.11 − 1.32i)67-s + (0.939 + 0.342i)73-s + (−0.592 + 1.62i)79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9328506062\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9328506062\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 5 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.592 + 1.62i)T + (-0.766 + 0.642i)T^{2} \) |
| 17 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 23 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 29 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 31 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + 1.73iT - T^{2} \) |
| 41 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 43 | \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 47 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 53 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 59 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 61 | \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 67 | \( 1 + (-1.11 + 1.32i)T + (-0.173 - 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 73 | \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 79 | \( 1 + (0.592 - 1.62i)T + (-0.766 - 0.642i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 97 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
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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
−8.751250659973276047048394011201, −7.82868596897361409563620551341, −7.04178510433641278623701960691, −6.21522395940928886370091841634, −5.42640658881532484441676041647, −5.21317530288160704702467077736, −3.85007157139903234475683604127, −2.77468861909666902686283658421, −2.08292730436954143049632272039, −0.54026252049718986815742804031,
1.58630910584208067982444714723, 2.75206563062015107551671573741, 3.58007628174006399173198172378, 4.30061595664893326973557129657, 5.03297270574686383578879685917, 6.40506211070238154745922164001, 6.98395873726460555490665727000, 7.33413592952202855610594428568, 8.342723872157764076271803523453, 9.010118110837426627796673779184