L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s − 0.999·8-s + 0.999·10-s + (1 + 1.73i)11-s + (0.5 − 0.866i)13-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s − 19-s + (0.499 + 0.866i)20-s + (−0.999 + 1.73i)22-s + (−0.5 + 0.866i)23-s + (−0.499 − 0.866i)25-s + 0.999·26-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s − 0.999·8-s + 0.999·10-s + (1 + 1.73i)11-s + (0.5 − 0.866i)13-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s − 19-s + (0.499 + 0.866i)20-s + (−0.999 + 1.73i)22-s + (−0.5 + 0.866i)23-s + (−0.499 − 0.866i)25-s + 0.999·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.804919997\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.804919997\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
good | 7 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - 2T + T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + 2T + T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−8.862382048326983612749630284985, −8.274814560525591601061423390884, −7.59676089410365672855070319435, −6.63287889688520218080751389765, −5.94346148391479568275326754781, −5.31123148712477599616525947491, −4.53022784230852668299061356383, −3.97443289757123498831039545828, −2.54605865091015352993352049611, −1.57527254847131054805278285823,
1.02918965750779873688277565407, 2.05325035246164575603023477542, 3.06567604083150391989256361889, 3.94559236196324303045140028273, 4.36605480183111432885798404319, 5.70620898377078290387090232166, 6.30395374696762146082758205902, 6.75934181824754414442003166572, 8.052336699540254873645857124222, 8.801166081160006413097852162282