| L(s) = 1 | − 5-s + 3·7-s + 3·11-s − 17-s + 8·19-s + 3·23-s + 25-s − 6·29-s + 10·31-s − 3·35-s + 37-s + 5·41-s − 6·43-s − 8·47-s + 2·49-s − 5·53-s − 3·55-s + 4·59-s + 13·61-s + 8·67-s + 11·71-s + 2·73-s + 9·77-s − 5·79-s − 6·83-s + 85-s + 15·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.13·7-s + 0.904·11-s − 0.242·17-s + 1.83·19-s + 0.625·23-s + 1/5·25-s − 1.11·29-s + 1.79·31-s − 0.507·35-s + 0.164·37-s + 0.780·41-s − 0.914·43-s − 1.16·47-s + 2/7·49-s − 0.686·53-s − 0.404·55-s + 0.520·59-s + 1.66·61-s + 0.977·67-s + 1.30·71-s + 0.234·73-s + 1.02·77-s − 0.562·79-s − 0.658·83-s + 0.108·85-s + 1.58·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.499304902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.499304902\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 11 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−14.28095525032996, −14.04269782608694, −13.20588598684067, −12.90055239967396, −12.00664105320345, −11.73993005736303, −11.19444284600844, −11.15805108186507, −10.04095336722486, −9.768658683208712, −9.120283007614312, −8.577302086683398, −7.983569546111789, −7.691463553138643, −6.952099641205001, −6.562881032017509, −5.771624007952504, −5.003489877171573, −4.876176650057337, −4.003727720254182, −3.517045442127686, −2.823514441147701, −1.976015202935645, −1.249134126064448, −0.7215747620275868,
0.7215747620275868, 1.249134126064448, 1.976015202935645, 2.823514441147701, 3.517045442127686, 4.003727720254182, 4.876176650057337, 5.003489877171573, 5.771624007952504, 6.562881032017509, 6.952099641205001, 7.691463553138643, 7.983569546111789, 8.577302086683398, 9.120283007614312, 9.768658683208712, 10.04095336722486, 11.15805108186507, 11.19444284600844, 11.73993005736303, 12.00664105320345, 12.90055239967396, 13.20588598684067, 14.04269782608694, 14.28095525032996
