L(s) = 1 | + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (−1.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s − 0.999·12-s + 13-s + (−0.499 − 0.866i)16-s + (−1.5 − 0.866i)19-s + 1.73i·21-s + 0.999·27-s + (−1.5 + 0.866i)28-s + (0.499 + 0.866i)36-s + (−0.5 − 0.866i)39-s + (−0.5 + 0.866i)43-s + (−0.499 + 0.866i)48-s + (1 + 1.73i)49-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (−1.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s − 0.999·12-s + 13-s + (−0.499 − 0.866i)16-s + (−1.5 − 0.866i)19-s + 1.73i·21-s + 0.999·27-s + (−1.5 + 0.866i)28-s + (0.499 + 0.866i)36-s + (−0.5 − 0.866i)39-s + (−0.5 + 0.866i)43-s + (−0.499 + 0.866i)48-s + (1 + 1.73i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6832076425\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6832076425\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + 1.73iT - T^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−10.04375524246975758218060622369, −9.152383049186592932706878777873, −8.032530829458682620797121120378, −6.87061299290143230781790529117, −6.55207435947450918821557432695, −5.94612061629038566264218035499, −4.71699370965639685397361418056, −3.35510042825608865533808537572, −2.08624037924958537313003745850, −0.66061533767724240031623338626,
2.44510462293072115045458923242, 3.50484480480085403486912670265, 4.04866188662035645577902502127, 5.60152252350275321508479302945, 6.27159936783007483939384332111, 6.87816176147505392195646336754, 8.426514500827832571216009803274, 8.791037834137066813027696600432, 9.818694976065765683003596648741, 10.51765468841867645645644145286