| L(s) = 1 | − 3-s − 5-s + 2.56·7-s + 9-s + 5.12·11-s + 2·13-s + 15-s + 4.56·17-s − 1.12·19-s − 2.56·21-s + 23-s + 25-s − 27-s + 8.56·29-s + 3.68·31-s − 5.12·33-s − 2.56·35-s + 0.561·37-s − 2·39-s + 3.43·41-s + 6.24·43-s − 45-s − 8·47-s − 0.438·49-s − 4.56·51-s + 0.561·53-s − 5.12·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.968·7-s + 0.333·9-s + 1.54·11-s + 0.554·13-s + 0.258·15-s + 1.10·17-s − 0.257·19-s − 0.558·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s + 1.58·29-s + 0.661·31-s − 0.891·33-s − 0.432·35-s + 0.0923·37-s − 0.320·39-s + 0.536·41-s + 0.952·43-s − 0.149·45-s − 1.16·47-s − 0.0626·49-s − 0.638·51-s + 0.0771·53-s − 0.690·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.192694571\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.192694571\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 - 5.12T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 4.56T + 17T^{2} \) |
| 19 | \( 1 + 1.12T + 19T^{2} \) |
| 29 | \( 1 - 8.56T + 29T^{2} \) |
| 31 | \( 1 - 3.68T + 31T^{2} \) |
| 37 | \( 1 - 0.561T + 37T^{2} \) |
| 41 | \( 1 - 3.43T + 41T^{2} \) |
| 43 | \( 1 - 6.24T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 0.561T + 53T^{2} \) |
| 59 | \( 1 + 6.56T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 - 2.56T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 0.315T + 83T^{2} \) |
| 89 | \( 1 + 3.12T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−8.215450934139831837003691124136, −7.43041125398753764532457731840, −6.64000514626343515321243053579, −6.09816665520392107885204761181, −5.21321340797861564624400875713, −4.44386113429387567014438307518, −3.91471208576146672670241265459, −2.90298709815738285260474802362, −1.50309366246724326555760133871, −0.942570827195454114134454540525,
0.942570827195454114134454540525, 1.50309366246724326555760133871, 2.90298709815738285260474802362, 3.91471208576146672670241265459, 4.44386113429387567014438307518, 5.21321340797861564624400875713, 6.09816665520392107885204761181, 6.64000514626343515321243053579, 7.43041125398753764532457731840, 8.215450934139831837003691124136
