| 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
−10.04639285940001059266314883186, −9.407369038532345528920324002771, −8.882899649259346655214099451736, −7.85522315719532557167639211720, −6.62977228105869773339911850629, −6.04176238074487916312449381591, −5.18413098612110981342046650999, −3.78291158537856948550184766188, −2.96074921698138298815554395451, −1.42256470701276481274939753294,
0.66995764621048348264482777424, 2.58361366427063090311032775481, 3.52156729761098020417963225444, 4.46260031546184812037712344365, 5.96342774466548734388054680414, 6.49402092585909944256212709906, 7.29985441555685171401466619185, 8.321080436621838151683983134135, 9.389020159457815646698700554285, 10.07993474760824511440999523637
