| L(s) = 1 | + 3·3-s − 7-s + 6·9-s − 4·11-s − 13-s + 7·17-s + 19-s − 3·21-s − 5·23-s + 9·27-s + 7·29-s + 2·31-s − 12·33-s + 6·37-s − 3·39-s + 6·41-s + 10·43-s − 8·47-s − 6·49-s + 21·51-s + 3·53-s + 3·57-s − 5·59-s − 8·61-s − 6·63-s + 11·67-s − 15·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.377·7-s + 2·9-s − 1.20·11-s − 0.277·13-s + 1.69·17-s + 0.229·19-s − 0.654·21-s − 1.04·23-s + 1.73·27-s + 1.29·29-s + 0.359·31-s − 2.08·33-s + 0.986·37-s − 0.480·39-s + 0.937·41-s + 1.52·43-s − 1.16·47-s − 6/7·49-s + 2.94·51-s + 0.412·53-s + 0.397·57-s − 0.650·59-s − 1.02·61-s − 0.755·63-s + 1.34·67-s − 1.80·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.884546023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.884546023\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−7.888899081299124096515659329767, −7.65192253480053974974436261823, −6.64251944739857158704235325346, −5.80051358432123065970718777777, −4.92463440591419303813451788372, −4.10877965808413861145858959017, −3.27881677317138417040673959605, −2.79544170860072098636734884985, −2.11420026291854542345489997201, −0.913394456320598610277665652276,
0.913394456320598610277665652276, 2.11420026291854542345489997201, 2.79544170860072098636734884985, 3.27881677317138417040673959605, 4.10877965808413861145858959017, 4.92463440591419303813451788372, 5.80051358432123065970718777777, 6.64251944739857158704235325346, 7.65192253480053974974436261823, 7.888899081299124096515659329767
