| L(s) = 1 | + 4·7-s − 4·11-s + 13-s + 6·17-s + 4·23-s − 6·29-s + 8·31-s − 2·37-s − 10·41-s + 4·43-s − 8·47-s + 9·49-s + 2·53-s − 4·59-s − 14·61-s + 12·67-s − 8·71-s + 10·73-s − 16·77-s − 4·83-s − 10·89-s + 4·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 1.20·11-s + 0.277·13-s + 1.45·17-s + 0.834·23-s − 1.11·29-s + 1.43·31-s − 0.328·37-s − 1.56·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s + 0.274·53-s − 0.520·59-s − 1.79·61-s + 1.46·67-s − 0.949·71-s + 1.17·73-s − 1.82·77-s − 0.439·83-s − 1.05·89-s + 0.419·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.997473912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.997473912\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−13.18010700116537, −12.57070891645335, −12.20891584954403, −11.65294732915601, −11.21141403341226, −10.84602236395437, −10.34398069493100, −9.901592370863491, −9.393522531830230, −8.624366625952575, −8.288941281187847, −7.917752592162068, −7.477503335024156, −7.021267237850207, −6.235402070347608, −5.691200899005314, −5.112459051598173, −4.977446847444217, −4.340242996321007, −3.561076351074804, −3.071059537471554, −2.476972792651556, −1.686806487278740, −1.328166594766732, −0.5011023855468033,
0.5011023855468033, 1.328166594766732, 1.686806487278740, 2.476972792651556, 3.071059537471554, 3.561076351074804, 4.340242996321007, 4.977446847444217, 5.112459051598173, 5.691200899005314, 6.235402070347608, 7.021267237850207, 7.477503335024156, 7.917752592162068, 8.288941281187847, 8.624366625952575, 9.393522531830230, 9.901592370863491, 10.34398069493100, 10.84602236395437, 11.21141403341226, 11.65294732915601, 12.20891584954403, 12.57070891645335, 13.18010700116537
