L(s) = 1 | + 1.41i·2-s − 2.00·4-s − 2i·5-s − 2.82i·8-s + 2.82·10-s − 7.07·13-s + 4.00·16-s + 2i·17-s + 4.00i·20-s + 25-s − 10.0i·26-s + 9.89i·29-s + 5.65i·32-s − 2.82·34-s + 12·37-s + ⋯ |
L(s) = 1 | + 0.999i·2-s − 1.00·4-s − 0.894i·5-s − 1.00i·8-s + 0.894·10-s − 1.96·13-s + 1.00·16-s + 0.485i·17-s + 0.894i·20-s + 0.200·25-s − 1.96i·26-s + 1.83i·29-s + 1.00i·32-s − 0.485·34-s + 1.97·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9849348543\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9849348543\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 7.07T + 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 9.89iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 12T + 37T^{2} \) |
| 41 | \( 1 - 8iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 12.7iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 15.5T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 7.07T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 16iT - 89T^{2} \) |
| 97 | \( 1 + 18.3T + 97T^{2} \) |
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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
−9.449298616183838917414621344527, −8.684488491919867481334173091958, −7.934019858196403446279335392555, −7.25688583069308497763668962420, −6.44981964668487474632504286682, −5.38247457313729314879864001835, −4.87907470592395728979913472387, −4.12521246856687554544636401912, −2.74440393467039269855684966733, −1.09483860997592126830343195965,
0.42870500871592043268282414520, 2.29555633457514920523112979363, 2.66745170882070923274634382903, 3.85288770084505224453031272397, 4.73719201562354593869658620435, 5.56981111621362003000498615326, 6.73095876756720919594703733970, 7.53430224264892804657960042838, 8.259602376710048630558562084905, 9.497456836513005277281501049822