| L(s) = 1 | + 2i·3-s − 4i·7-s − 9-s − 11-s + 4i·13-s + 4·19-s + 8·21-s + 6i·23-s + 4i·27-s + 6·29-s + 8·31-s − 2i·33-s + 2i·37-s − 8·39-s + 6·41-s + ⋯ |
| L(s) = 1 | + 1.15i·3-s − 1.51i·7-s − 0.333·9-s − 0.301·11-s + 1.10i·13-s + 0.917·19-s + 1.74·21-s + 1.25i·23-s + 0.769i·27-s + 1.11·29-s + 1.43·31-s − 0.348i·33-s + 0.328i·37-s − 1.28·39-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.648016195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.648016195\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 7 | \( 1 + 4iT - 7T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 16iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−9.904619601947231462797622914774, −9.554126878613921355380491692879, −8.450224077549487878662739886029, −7.41231500998221510336887564749, −6.82407253323772120423210213616, −5.53452081378025250626141060079, −4.48517842398633277794619898587, −4.08321711411749834182680975996, −3.03417605940130255093621577693, −1.21163412986945572149062512868,
0.891608224610582192073426463959, 2.35617466104163218299710698300, 2.94430395547598527273179454973, 4.68887016627501765658936316281, 5.66435842287262031346413980528, 6.29833131786111481222644385444, 7.22756490142002398012909377944, 8.204583419709775097730619486966, 8.514919887081907299396917559478, 9.715580398034927405479650009820
