L(s) = 1 | + (0.565 + 0.978i)5-s + (−3.71 − 2.14i)7-s + (1.00 + 0.582i)11-s + (2.64 − 1.52i)13-s − 1.49i·17-s + 3.42·19-s + (−3.85 − 6.68i)23-s + (1.86 − 3.22i)25-s + (0.709 − 1.22i)29-s + (4.66 − 2.69i)31-s − 4.84i·35-s − 2.97i·37-s + (4.23 − 2.44i)41-s + (1.74 − 3.01i)43-s + (−1.77 + 3.08i)47-s + ⋯ |
L(s) = 1 | + (0.252 + 0.437i)5-s + (−1.40 − 0.810i)7-s + (0.304 + 0.175i)11-s + (0.733 − 0.423i)13-s − 0.362i·17-s + 0.785·19-s + (−0.804 − 1.39i)23-s + (0.372 − 0.644i)25-s + (0.131 − 0.228i)29-s + (0.837 − 0.483i)31-s − 0.819i·35-s − 0.488i·37-s + (0.661 − 0.381i)41-s + (0.265 − 0.460i)43-s + (−0.259 + 0.449i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.09128 - 0.679183i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.09128 - 0.679183i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.565 - 0.978i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (3.71 + 2.14i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.00 - 0.582i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.64 + 1.52i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 1.49iT - 17T^{2} \) |
| 19 | \( 1 - 3.42T + 19T^{2} \) |
| 23 | \( 1 + (3.85 + 6.68i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.709 + 1.22i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.66 + 2.69i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 2.97iT - 37T^{2} \) |
| 41 | \( 1 + (-4.23 + 2.44i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.74 + 3.01i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.77 - 3.08i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + (-7.50 + 4.33i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.16 + 1.82i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.58 - 9.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2.54T + 71T^{2} \) |
| 73 | \( 1 + 7.06T + 73T^{2} \) |
| 79 | \( 1 + (-2.24 - 1.29i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.98 + 2.30i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 8.63iT - 89T^{2} \) |
| 97 | \( 1 + (-3.35 + 5.81i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.07993474760824511440999523637, −9.389020159457815646698700554285, −8.321080436621838151683983134135, −7.29985441555685171401466619185, −6.49402092585909944256212709906, −5.96342774466548734388054680414, −4.46260031546184812037712344365, −3.52156729761098020417963225444, −2.58361366427063090311032775481, −0.66995764621048348264482777424,
1.42256470701276481274939753294, 2.96074921698138298815554395451, 3.78291158537856948550184766188, 5.18413098612110981342046650999, 6.04176238074487916312449381591, 6.62977228105869773339911850629, 7.85522315719532557167639211720, 8.882899649259346655214099451736, 9.407369038532345528920324002771, 10.04639285940001059266314883186