| L(s) = 1 | − 3-s + 9-s + 3·11-s + 13-s − 4·17-s + 19-s − 5·23-s − 27-s + 4·29-s − 2·31-s − 3·33-s + 3·37-s − 39-s − 3·41-s + 4·43-s − 9·47-s + 4·51-s − 3·53-s − 57-s + 12·59-s − 4·67-s + 5·69-s − 10·71-s + 2·73-s + 81-s + 8·83-s − 4·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.904·11-s + 0.277·13-s − 0.970·17-s + 0.229·19-s − 1.04·23-s − 0.192·27-s + 0.742·29-s − 0.359·31-s − 0.522·33-s + 0.493·37-s − 0.160·39-s − 0.468·41-s + 0.609·43-s − 1.31·47-s + 0.560·51-s − 0.412·53-s − 0.132·57-s + 1.56·59-s − 0.488·67-s + 0.601·69-s − 1.18·71-s + 0.234·73-s + 1/9·81-s + 0.878·83-s − 0.428·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−13.81883874228805, −13.28255033568370, −12.89762549936853, −12.25911420158841, −11.85716805656031, −11.41784916732117, −11.07889720105482, −10.37658403590505, −10.02958401787697, −9.449369406188386, −8.924570768226757, −8.499029568908559, −7.850136698193558, −7.349570553365510, −6.614053498796501, −6.414744767112616, −5.896543266162414, −5.201639500637838, −4.676714590180890, −4.098414980558068, −3.687327592437887, −2.907959933596193, −2.136283668981326, −1.560417829729421, −0.8284799084388063, 0,
0.8284799084388063, 1.560417829729421, 2.136283668981326, 2.907959933596193, 3.687327592437887, 4.098414980558068, 4.676714590180890, 5.201639500637838, 5.896543266162414, 6.414744767112616, 6.614053498796501, 7.349570553365510, 7.850136698193558, 8.499029568908559, 8.924570768226757, 9.449369406188386, 10.02958401787697, 10.37658403590505, 11.07889720105482, 11.41784916732117, 11.85716805656031, 12.25911420158841, 12.89762549936853, 13.28255033568370, 13.81883874228805
