L(s) = 1 | − 1.41i·2-s + 3.00i·3-s − 2.00·4-s + (3.95 + 3.05i)5-s + 4.25·6-s − 7.53·7-s + 2.82i·8-s − 0.0375·9-s + (4.31 − 5.59i)10-s − 2.07i·11-s − 6.01i·12-s + 19.5i·13-s + 10.6i·14-s + (−9.18 + 11.9i)15-s + 4.00·16-s − 22.4·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 1.00i·3-s − 0.500·4-s + (0.791 + 0.610i)5-s + 0.708·6-s − 1.07·7-s + 0.353i·8-s − 0.00417·9-s + (0.431 − 0.559i)10-s − 0.189i·11-s − 0.501i·12-s + 1.50i·13-s + 0.761i·14-s + (−0.612 + 0.793i)15-s + 0.250·16-s − 1.31·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0257 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0257 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.897854 + 0.875053i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.897854 + 0.875053i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 5 | \( 1 + (-3.95 - 3.05i)T \) |
| 23 | \( 1 + (18.5 - 13.5i)T \) |
good | 3 | \( 1 - 3.00iT - 9T^{2} \) |
| 7 | \( 1 + 7.53T + 49T^{2} \) |
| 11 | \( 1 + 2.07iT - 121T^{2} \) |
| 13 | \( 1 - 19.5iT - 169T^{2} \) |
| 17 | \( 1 + 22.4T + 289T^{2} \) |
| 19 | \( 1 - 7.65iT - 361T^{2} \) |
| 29 | \( 1 - 36.7T + 841T^{2} \) |
| 31 | \( 1 + 9.27T + 961T^{2} \) |
| 37 | \( 1 - 8.26T + 1.36e3T^{2} \) |
| 41 | \( 1 - 29.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 48.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 77.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 39.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + 19.3T + 3.48e3T^{2} \) |
| 61 | \( 1 - 11.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 23.2T + 4.48e3T^{2} \) |
| 71 | \( 1 + 9.60T + 5.04e3T^{2} \) |
| 73 | \( 1 + 28.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 88.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 70.8T + 6.88e3T^{2} \) |
| 89 | \( 1 - 130. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 169.T + 9.40e3T^{2} \) |
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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
−12.05756935702375912792356143074, −10.99580235293286170757341696806, −10.23668359458833629928292369975, −9.520781439720731365118358683944, −8.942739357825122235479904923517, −6.96542491738954440924292042662, −6.01583224481104252523477665117, −4.50459730674665409719567495340, −3.53649000667236811769382691657, −2.15184616569000785772427900700,
0.67228152665357585385279443463, 2.55400998704332161844517753946, 4.52992057731903793075029990082, 5.95529018139115055903238665694, 6.51308358597456688297527742319, 7.62134565796695192728555803177, 8.644791856084205800516513511356, 9.636741224134554441203985466165, 10.53136844972825719195622877355, 12.34995945235809208877044201356