| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 4·7-s − 8-s + 9-s − 10-s − 12-s − 13-s + 4·14-s − 15-s + 16-s + 2·17-s − 18-s + 20-s + 4·21-s − 4·23-s + 24-s + 25-s + 26-s − 27-s − 4·28-s − 10·29-s + 30-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.277·13-s + 1.06·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.223·20-s + 0.872·21-s − 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.755·28-s − 1.85·29-s + 0.182·30-s − 0.176·32-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 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.92179277166721, −14.44138512607509, −13.68965278063926, −13.12869593602404, −12.78601612963843, −12.31131468466164, −11.60622060644365, −11.28659577253237, −10.50243053783872, −10.02428905913417, −9.678645463953093, −9.385070916904966, −8.596864401320103, −7.994445515833397, −7.298190885364399, −6.856021680608059, −6.301752255553938, −5.827498574553483, −5.358555824297126, −4.508780015184209, −3.562412468350297, −3.291925033240161, −2.292246123568283, −1.752589190319034, −0.7032665511454389, 0,
0.7032665511454389, 1.752589190319034, 2.292246123568283, 3.291925033240161, 3.562412468350297, 4.508780015184209, 5.358555824297126, 5.827498574553483, 6.301752255553938, 6.856021680608059, 7.298190885364399, 7.994445515833397, 8.596864401320103, 9.385070916904966, 9.678645463953093, 10.02428905913417, 10.50243053783872, 11.28659577253237, 11.60622060644365, 12.31131468466164, 12.78601612963843, 13.12869593602404, 13.68965278063926, 14.44138512607509, 14.92179277166721
