L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−2.24 − 1.39i)7-s − 0.999i·8-s + (1.37 − 0.796i)11-s + 0.925i·13-s + (1.24 + 2.33i)14-s + (−0.5 + 0.866i)16-s + (1.97 + 3.42i)17-s + (−0.541 − 0.312i)19-s − 1.59·22-s + (−6.74 − 3.89i)23-s + (0.462 − 0.801i)26-s + (0.0890 − 2.64i)28-s + 9.34i·29-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.848 − 0.528i)7-s − 0.353i·8-s + (0.415 − 0.240i)11-s + 0.256i·13-s + (0.332 + 0.623i)14-s + (−0.125 + 0.216i)16-s + (0.479 + 0.831i)17-s + (−0.124 − 0.0717i)19-s − 0.339·22-s + (−1.40 − 0.811i)23-s + (0.0907 − 0.157i)26-s + (0.0168 − 0.499i)28-s + 1.73i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7869755927\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7869755927\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.24 + 1.39i)T \) |
good | 11 | \( 1 + (-1.37 + 0.796i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 0.925iT - 13T^{2} \) |
| 17 | \( 1 + (-1.97 - 3.42i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.541 + 0.312i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (6.74 + 3.89i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 9.34iT - 29T^{2} \) |
| 31 | \( 1 + (-8.94 + 5.16i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.213 - 0.369i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 8.35T + 41T^{2} \) |
| 43 | \( 1 + 6.27T + 43T^{2} \) |
| 47 | \( 1 + (1.39 - 2.40i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.90 - 1.67i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.10 + 5.38i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-9.52 - 5.50i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.178 - 0.308i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 9.07iT - 71T^{2} \) |
| 73 | \( 1 + (-5.91 + 3.41i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.52 + 7.83i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 0.809T + 83T^{2} \) |
| 89 | \( 1 + (2.00 - 3.47i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 7.87iT - 97T^{2} \) |
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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
−8.707646229340284736895282445318, −8.267255191855166654062890874815, −7.36636992716393573730113771008, −6.50110302121535567941376699095, −6.14748322090167156194785859207, −4.80474594541315235136142232940, −3.84317013103081627319001619529, −3.25453379199927176337296928766, −2.09966350027762316649076791840, −0.971630172883152419548890067614,
0.35739825557514583431877341096, 1.78697066530437441568778080714, 2.82512595225820125023508987380, 3.73604385739763744654839646187, 4.88066502129699075544300699441, 5.72802167430087252222100277221, 6.41515895432408134543492577626, 7.00046420994229076714313948392, 8.009111539579399325583136295526, 8.421037360299255376474633865984