| L(s) = 1 | − 2·7-s − 2·11-s − 13-s + 2·17-s − 4·19-s + 23-s + 29-s − 2·31-s − 12·41-s − 5·43-s + 4·47-s − 3·49-s + 9·53-s + 8·59-s − 7·61-s + 14·67-s + 6·73-s + 4·77-s − 15·79-s + 4·83-s + 18·89-s + 2·91-s − 16·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 0.603·11-s − 0.277·13-s + 0.485·17-s − 0.917·19-s + 0.208·23-s + 0.185·29-s − 0.359·31-s − 1.87·41-s − 0.762·43-s + 0.583·47-s − 3/7·49-s + 1.23·53-s + 1.04·59-s − 0.896·61-s + 1.71·67-s + 0.702·73-s + 0.455·77-s − 1.68·79-s + 0.439·83-s + 1.90·89-s + 0.209·91-s − 1.62·97-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
| \(L(1)\) |
\(\approx\) |
\(0.6215002809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6215002809\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−13.24417597872941, −12.64568932501028, −12.21751399708685, −11.84934385172913, −11.20075628774791, −10.67455853046377, −10.30218803804400, −9.822796465728685, −9.469780177365289, −8.763565899120469, −8.375949219204326, −7.932777236113420, −7.291881435791989, −6.698300834170504, −6.560868409737031, −5.748101590274393, −5.253453856357517, −4.926310754184108, −4.020222227518237, −3.732208356220256, −2.994053688738861, −2.557984776614426, −1.903460170066903, −1.156551068387448, −0.2312848340615203,
0.2312848340615203, 1.156551068387448, 1.903460170066903, 2.557984776614426, 2.994053688738861, 3.732208356220256, 4.020222227518237, 4.926310754184108, 5.253453856357517, 5.748101590274393, 6.560868409737031, 6.698300834170504, 7.291881435791989, 7.932777236113420, 8.375949219204326, 8.763565899120469, 9.469780177365289, 9.822796465728685, 10.30218803804400, 10.67455853046377, 11.20075628774791, 11.84934385172913, 12.21751399708685, 12.64568932501028, 13.24417597872941
