L(s) = 1 | + (1.28 + 0.742i)2-s + (0.101 + 0.176i)4-s + (−0.154 − 0.267i)5-s − 2.66i·8-s − 0.457i·10-s + (2.73 + 1.58i)11-s + (−3.00 + 1.73i)13-s + (2.18 − 3.78i)16-s + 4.88·17-s − 5.34i·19-s + (0.0314 − 0.0544i)20-s + (2.34 + 4.06i)22-s + (5.17 − 2.98i)23-s + (2.45 − 4.24i)25-s − 5.14·26-s + ⋯ |
L(s) = 1 | + (0.909 + 0.524i)2-s + (0.0509 + 0.0882i)4-s + (−0.0689 − 0.119i)5-s − 0.942i·8-s − 0.144i·10-s + (0.825 + 0.476i)11-s + (−0.833 + 0.481i)13-s + (0.545 − 0.945i)16-s + 1.18·17-s − 1.22i·19-s + (0.00702 − 0.0121i)20-s + (0.500 + 0.866i)22-s + (1.07 − 0.622i)23-s + (0.490 − 0.849i)25-s − 1.00·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.240i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.613338733\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.613338733\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1.28 - 0.742i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (0.154 + 0.267i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.73 - 1.58i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.00 - 1.73i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 4.88T + 17T^{2} \) |
| 19 | \( 1 + 5.34iT - 19T^{2} \) |
| 23 | \( 1 + (-5.17 + 2.98i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.70 + 1.56i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (6.51 - 3.76i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 11.8T + 37T^{2} \) |
| 41 | \( 1 + (-2.58 - 4.48i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.75 + 4.76i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.23 + 7.33i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 0.0855iT - 53T^{2} \) |
| 59 | \( 1 + (1.04 + 1.80i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.69 + 2.71i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.0554 - 0.0959i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 7.78iT - 71T^{2} \) |
| 73 | \( 1 + 9.61iT - 73T^{2} \) |
| 79 | \( 1 + (2.56 - 4.44i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.42 - 7.66i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 1.87T + 89T^{2} \) |
| 97 | \( 1 + (-10.9 - 6.34i)T + (48.5 + 84.0i)T^{2} \) |
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
−9.455113244435777043000276231236, −8.991488300668847225241725606285, −7.61682765649367752812443575388, −6.97731735877839983029957288515, −6.29553828879121524241430961906, −5.23475348444713075202057651274, −4.64135385711149455635558743076, −3.81551926352206751262562780376, −2.59185548841430274959968205643, −0.931759569615190447775808498419,
1.39566569036879101433603460832, 2.83456219912169189073386561951, 3.51627263972537898981684579374, 4.36045907401379233901994730000, 5.47750408209925692512809595496, 5.91069743767161976013909405447, 7.41209512104664298835978758247, 7.82228353066982609290607603270, 9.017146614779151713420928036818, 9.633023022529432590691017880664