| L(s) = 1 | + 2·7-s − 11-s − 2·13-s + 2·19-s + 23-s − 5·25-s − 10·29-s − 4·31-s − 2·37-s + 2·41-s + 2·43-s − 8·47-s − 3·49-s − 4·53-s + 12·59-s + 6·61-s + 2·67-s − 6·73-s − 2·77-s − 2·79-s − 4·83-s + 8·89-s − 4·91-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.301·11-s − 0.554·13-s + 0.458·19-s + 0.208·23-s − 25-s − 1.85·29-s − 0.718·31-s − 0.328·37-s + 0.312·41-s + 0.304·43-s − 1.16·47-s − 3/7·49-s − 0.549·53-s + 1.56·59-s + 0.768·61-s + 0.244·67-s − 0.702·73-s − 0.227·77-s − 0.225·79-s − 0.439·83-s + 0.847·89-s − 0.419·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.315033970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.315033970\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.34454920496125, −12.79276174058355, −12.62038669677243, −11.68232326491424, −11.51895048847625, −11.13422759452138, −10.51573816134724, −9.929749077435625, −9.580015818427534, −9.057036640440434, −8.482838620398161, −7.904392124954620, −7.578205371793785, −7.123447642112862, −6.514646648542286, −5.753375905241735, −5.390320546699819, −4.980669241073074, −4.283440339093491, −3.749710759018180, −3.205468504221338, −2.389319437821459, −1.905325968516967, −1.339317495740677, −0.3309215769270298,
0.3309215769270298, 1.339317495740677, 1.905325968516967, 2.389319437821459, 3.205468504221338, 3.749710759018180, 4.283440339093491, 4.980669241073074, 5.390320546699819, 5.753375905241735, 6.514646648542286, 7.123447642112862, 7.578205371793785, 7.904392124954620, 8.482838620398161, 9.057036640440434, 9.580015818427534, 9.929749077435625, 10.51573816134724, 11.13422759452138, 11.51895048847625, 11.68232326491424, 12.62038669677243, 12.79276174058355, 13.34454920496125
