| L(s) = 1 | + 3.46i·5-s − 2.64·7-s + 4.58i·13-s + 1.73i·17-s − 5.29·19-s − 4.58i·23-s − 6.99·25-s − 7.93·29-s − 2.64·31-s − 9.16i·35-s + 4·37-s − 6.92i·41-s − 1.73i·43-s + 6·47-s + 7.00·49-s + ⋯ |
| L(s) = 1 | + 1.54i·5-s − 0.999·7-s + 1.27i·13-s + 0.420i·17-s − 1.21·19-s − 0.955i·23-s − 1.39·25-s − 1.47·29-s − 0.475·31-s − 1.54i·35-s + 0.657·37-s − 1.08i·41-s − 0.264i·43-s + 0.875·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1402837682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1402837682\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
| good | 5 | \( 1 - 3.46iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 4.58iT - 13T^{2} \) |
| 17 | \( 1 - 1.73iT - 17T^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 23 | \( 1 + 4.58iT - 23T^{2} \) |
| 29 | \( 1 + 7.93T + 29T^{2} \) |
| 31 | \( 1 + 2.64T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 6.92iT - 41T^{2} \) |
| 43 | \( 1 + 1.73iT - 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 7.93T + 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 - 9.16iT - 61T^{2} \) |
| 67 | \( 1 - 12.1iT - 67T^{2} \) |
| 71 | \( 1 + 4.58iT - 71T^{2} \) |
| 73 | \( 1 + 9.16iT - 73T^{2} \) |
| 79 | \( 1 + 6.92iT - 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 5.19iT - 89T^{2} \) |
| 97 | \( 1 + 9.16iT - 97T^{2} \) |
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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
−9.174648629453894644998907135047, −8.646411670861863688751205814590, −7.33331405619968924022660513484, −7.02076229454800964088787025039, −6.27994315219444186500219944890, −5.76515481745639191221129393818, −4.20501898254237431183322129056, −3.76370380319991488130551559939, −2.68029201200092424888316633124, −2.04106665427616184085299894784,
0.04664815682414418271997626973, 1.09622869285781823564288051050, 2.39462478219914622774970085267, 3.53012362507592307576924325945, 4.25992523959258297305155712626, 5.31975209763907443802360889622, 5.68676832819156409439262995130, 6.66056950412088264666301370336, 7.67316174394703813250906442950, 8.223588405683875632275266182154
