| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 2·7-s − 8-s + 9-s + 10-s − 12-s + 13-s + 2·14-s + 15-s + 16-s + 4·17-s − 18-s + 2·19-s − 20-s + 2·21-s − 6·23-s + 24-s + 25-s − 26-s − 27-s − 2·28-s + 4·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.277·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.458·19-s − 0.223·20-s + 0.436·21-s − 1.25·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.377·28-s + 0.742·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.81199506002496, −14.55036004735304, −13.68658754307689, −13.33810012937449, −12.43814342905393, −12.30041315392268, −11.67898559097719, −11.32225213081628, −10.52465321561459, −10.07891461592279, −9.892465834565188, −9.063556733228209, −8.548913063335464, −8.008541786662589, −7.374449381684429, −6.926114828039224, −6.391527389700145, −5.658007569329412, −5.382894537182903, −4.401394187804597, −3.697700006651786, −3.277448814034030, −2.410303851634197, −1.553095447870013, −0.7741015905013478, 0,
0.7741015905013478, 1.553095447870013, 2.410303851634197, 3.277448814034030, 3.697700006651786, 4.401394187804597, 5.382894537182903, 5.658007569329412, 6.391527389700145, 6.926114828039224, 7.374449381684429, 8.008541786662589, 8.548913063335464, 9.063556733228209, 9.892465834565188, 10.07891461592279, 10.52465321561459, 11.32225213081628, 11.67898559097719, 12.30041315392268, 12.43814342905393, 13.33810012937449, 13.68658754307689, 14.55036004735304, 14.81199506002496
