| L(s) = 1 | − i·7-s − 6·11-s + i·13-s + 19-s + 6i·23-s + 6·29-s + 8·31-s − 7i·37-s + 6·41-s + 4i·43-s − 12i·47-s + 6·49-s − 6i·53-s + 11·61-s − 7i·67-s + ⋯ |
| L(s) = 1 | − 0.377i·7-s − 1.80·11-s + 0.277i·13-s + 0.229·19-s + 1.25i·23-s + 1.11·29-s + 1.43·31-s − 1.15i·37-s + 0.937·41-s + 0.609i·43-s − 1.75i·47-s + 0.857·49-s − 0.824i·53-s + 1.40·61-s − 0.855i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.495777555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.495777555\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 7iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 11T + 61T^{2} \) |
| 67 | \( 1 + 7iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 11iT - 73T^{2} \) |
| 79 | \( 1 - T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 + 13iT - 97T^{2} \) |
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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
−8.662005344780482857512107138103, −7.950961017954233406849241021547, −7.40183345547681027639334664554, −6.54642944371342397416588372841, −5.54597927496260557904653656806, −4.98562726913011504771304098781, −4.00440381354498784722132274492, −2.99734575086180618444286341179, −2.14203024117508226005110013508, −0.65951577567304629641748861218,
0.854476625427570910611834974982, 2.57853459821101907334827209308, 2.80069967900779266725902995997, 4.30199016195112327218552354374, 4.99934506431491502306299490311, 5.76888189222842868587102579353, 6.55668090600298430594086124511, 7.50436432220991161132668150533, 8.228750818772929242373553556010, 8.646447319931435621723778486705
