L(s) = 1 | + 2.82i·5-s − 7-s + 3·9-s − 2.82i·11-s + 2.82i·13-s − 2·17-s + 5.65i·19-s + 8·23-s − 3.00·25-s + 5.65i·29-s − 8·31-s − 2.82i·35-s + 5.65i·37-s − 6·41-s − 2.82i·43-s + ⋯ |
L(s) = 1 | + 1.26i·5-s − 0.377·7-s + 9-s − 0.852i·11-s + 0.784i·13-s − 0.485·17-s + 1.29i·19-s + 1.66·23-s − 0.600·25-s + 1.05i·29-s − 1.43·31-s − 0.478i·35-s + 0.929i·37-s − 0.937·41-s − 0.431i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.01929 + 1.01929i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01929 + 1.01929i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 - 3T^{2} \) |
| 5 | \( 1 - 2.82iT - 5T^{2} \) |
| 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 - 2.82iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 5.65iT - 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 - 5.65iT - 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 5.65iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 2.82iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 11.3iT - 53T^{2} \) |
| 59 | \( 1 - 11.3iT - 59T^{2} \) |
| 61 | \( 1 + 2.82iT - 61T^{2} \) |
| 67 | \( 1 + 8.48iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 5.65iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−10.55693802604034433048460987565, −9.534353863920074951591938950128, −8.769216947924663779715726626624, −7.55037972790973719329599722494, −6.84998691710577821175779098970, −6.30463753630230482359099148310, −5.05104511984775916086924410195, −3.76252744201132026147857553808, −3.07547913068284548591010114276, −1.63363373935741372991009952438,
0.72093103111476239774625952082, 2.09994838413178607782110618471, 3.62489600639547292707234514649, 4.79330220502070610909203398861, 5.14410782731278676511379225704, 6.62895490476168478466049476046, 7.30343104806616568061480299580, 8.307575897326408052265656012030, 9.251161596018308489714508592453, 9.637362799388285792536137427651