| L(s) = 1 | − 7-s + 6·11-s + 6·13-s − 2·17-s − 7·19-s + 8·23-s − 6·29-s + 9·31-s + 3·37-s + 10·41-s + 43-s − 2·47-s − 6·49-s − 2·53-s + 12·59-s + 3·61-s + 4·67-s − 12·71-s − 11·73-s − 6·77-s + 11·79-s − 6·83-s + 8·89-s − 6·91-s − 7·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.80·11-s + 1.66·13-s − 0.485·17-s − 1.60·19-s + 1.66·23-s − 1.11·29-s + 1.61·31-s + 0.493·37-s + 1.56·41-s + 0.152·43-s − 0.291·47-s − 6/7·49-s − 0.274·53-s + 1.56·59-s + 0.384·61-s + 0.488·67-s − 1.42·71-s − 1.28·73-s − 0.683·77-s + 1.23·79-s − 0.658·83-s + 0.847·89-s − 0.628·91-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.599516528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.599516528\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−16.54588978516086, −16.00719551215696, −15.37483293576093, −14.70571861895140, −14.45241561476310, −13.51790829406565, −13.08278326490905, −12.70726571311394, −11.74927141797842, −11.25466427476879, −10.92893912706043, −10.08545065921322, −9.314244742784591, −8.817995290254964, −8.525308941466494, −7.524769935039645, −6.560195807100374, −6.473207236242927, −5.820993275067504, −4.661305397561465, −4.068667630826291, −3.550427510338673, −2.595644160559951, −1.541073198886355, −0.8172188875419737,
0.8172188875419737, 1.541073198886355, 2.595644160559951, 3.550427510338673, 4.068667630826291, 4.661305397561465, 5.820993275067504, 6.473207236242927, 6.560195807100374, 7.524769935039645, 8.525308941466494, 8.817995290254964, 9.314244742784591, 10.08545065921322, 10.92893912706043, 11.25466427476879, 11.74927141797842, 12.70726571311394, 13.08278326490905, 13.51790829406565, 14.45241561476310, 14.70571861895140, 15.37483293576093, 16.00719551215696, 16.54588978516086
