| L(s) = 1 | + (−1 + 2i)5-s − 4i·7-s − 11-s − 6i·13-s + 2i·17-s − 4·19-s − 6i·23-s + (−3 − 4i)25-s − 2·29-s + 8·31-s + (8 + 4i)35-s + 8i·37-s − 6·41-s + 12i·43-s + 10i·47-s + ⋯ |
| L(s) = 1 | + (−0.447 + 0.894i)5-s − 1.51i·7-s − 0.301·11-s − 1.66i·13-s + 0.485i·17-s − 0.917·19-s − 1.25i·23-s + (−0.600 − 0.800i)25-s − 0.371·29-s + 1.43·31-s + (1.35 + 0.676i)35-s + 1.31i·37-s − 0.937·41-s + 1.82i·43-s + 1.45i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1 - 2i)T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 - 10iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 - 4iT - 97T^{2} \) |
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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
−8.094005497106503424522686677155, −7.38423699253185673021647901859, −6.52940496700225220335212698709, −6.12650605190121258613463840327, −4.76669528937552966059545611232, −4.24087541192893065647347719401, −3.25374408795945736250518516221, −2.71120007214133344256004293538, −1.14268767624092829376826341943, 0,
1.69763026056002695589171623761, 2.34095725083537300450962399157, 3.58617630609164311183359368232, 4.41388260512417558592231944777, 5.18471647514324138086839513350, 5.76071479032182738164383061399, 6.67664847513532635198327944940, 7.46098434412460710736999700070, 8.390077121471355053278739533051
