L(s) = 1 | + (1.65 − 0.521i)3-s + 0.575i·5-s + 0.301i·7-s + (2.45 − 1.72i)9-s + 1.39·11-s + 2.30·13-s + (0.300 + 0.950i)15-s − i·17-s − 0.707i·19-s + (0.157 + 0.498i)21-s − 1.25·23-s + 4.66·25-s + (3.15 − 4.12i)27-s + 4.73i·29-s − 0.394i·31-s + ⋯ |
L(s) = 1 | + (0.953 − 0.301i)3-s + 0.257i·5-s + 0.114i·7-s + (0.818 − 0.574i)9-s + 0.419·11-s + 0.640·13-s + (0.0774 + 0.245i)15-s − 0.242i·17-s − 0.162i·19-s + (0.0343 + 0.108i)21-s − 0.260·23-s + 0.933·25-s + (0.607 − 0.794i)27-s + 0.878i·29-s − 0.0709i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.215i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.28347 - 0.249548i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.28347 - 0.249548i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.65 + 0.521i)T \) |
| 17 | \( 1 + iT \) |
good | 5 | \( 1 - 0.575iT - 5T^{2} \) |
| 7 | \( 1 - 0.301iT - 7T^{2} \) |
| 11 | \( 1 - 1.39T + 11T^{2} \) |
| 13 | \( 1 - 2.30T + 13T^{2} \) |
| 19 | \( 1 + 0.707iT - 19T^{2} \) |
| 23 | \( 1 + 1.25T + 23T^{2} \) |
| 29 | \( 1 - 4.73iT - 29T^{2} \) |
| 31 | \( 1 + 0.394iT - 31T^{2} \) |
| 37 | \( 1 - 4.79T + 37T^{2} \) |
| 41 | \( 1 - 4.30iT - 41T^{2} \) |
| 43 | \( 1 + 8.76iT - 43T^{2} \) |
| 47 | \( 1 + 5.72T + 47T^{2} \) |
| 53 | \( 1 - 7.42iT - 53T^{2} \) |
| 59 | \( 1 + 4.06T + 59T^{2} \) |
| 61 | \( 1 + 3.71T + 61T^{2} \) |
| 67 | \( 1 + 7.90iT - 67T^{2} \) |
| 71 | \( 1 + 8.95T + 71T^{2} \) |
| 73 | \( 1 + 5.50T + 73T^{2} \) |
| 79 | \( 1 + 0.301iT - 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 + 0.377iT - 89T^{2} \) |
| 97 | \( 1 - 2.69T + 97T^{2} \) |
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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
−10.12500663986407868822825368257, −9.130197157737378173525048270144, −8.661580049398216727938496139443, −7.64976532054416092273370325312, −6.88696808558630370119828283592, −6.02366800781236084010275112410, −4.63883244730762050649305018664, −3.58233530207901781680076542748, −2.67371866245664200586175206062, −1.34201998811088221511220100159,
1.42167948342030578257841050966, 2.77603546744637697596923118625, 3.85922843558193582066695814022, 4.62157994715546561647970938642, 5.89722624025740395193907193924, 6.93400200474700974788050104100, 7.933183056811685315491164000916, 8.591800815929445079933076438925, 9.348940935323661702884776110192, 10.12273763209341643510829275225