| L(s) = 1 | + i·3-s − 2·5-s + 2i·7-s − 9-s − 4i·11-s − 6i·13-s − 2i·15-s − 2·21-s − 25-s − i·27-s − 2i·29-s + 8·31-s + 4·33-s − 4i·35-s + 10·37-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.894·5-s + 0.755i·7-s − 0.333·9-s − 1.20i·11-s − 1.66i·13-s − 0.516i·15-s − 0.436·21-s − 0.200·25-s − 0.192i·27-s − 0.371i·29-s + 1.43·31-s + 0.696·33-s − 0.676i·35-s + 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.925985 - 0.445051i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.925985 - 0.445051i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 41 | \( 1 + (5 - 4i)T \) |
| good | 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 10iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 14iT - 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 + 6iT - 79T^{2} \) |
| 83 | \( 1 + 16T + 83T^{2} \) |
| 89 | \( 1 + 4iT - 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−9.973555473449045061689183005016, −8.912146789985222110613868421730, −8.211667857697359480242970875774, −7.70967088256427422503576667165, −6.18922722803950013603389748862, −5.59678728789606218694066264948, −4.54701466385807946061990401767, −3.44917782704188721746079885074, −2.75134261588590405774336849301, −0.52386765181596440161221694902,
1.30042500314380878635626878308, 2.61257360800506047324889181381, 4.22454791601923624547465509936, 4.39583393213549346749205940281, 6.03205645092757644053203528644, 7.08136493563765117926286496732, 7.34514788027834800483476918576, 8.299826622472109557705126003640, 9.279316026144734963263739618604, 10.07273894811391958392402133191
