| 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 + 6·17-s − 18-s − 4·19-s + 20-s − 4·21-s − 24-s + 25-s − 26-s + 27-s − 4·28-s + 8·29-s − 30-s + 8·31-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 + 1.45·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.872·21-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.755·28-s + 1.48·29-s − 0.182·30-s + 1.43·31-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
| \(L(1)\) |
\(\approx\) |
\(2.211860870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.211860870\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−14.54143168122536, −14.20239856527712, −13.38783351927800, −13.13940929925929, −12.59895758600131, −12.00644742699426, −11.58689596411431, −10.60908313975156, −10.22174525065697, −9.825181723872391, −9.560833308513423, −8.785432962502342, −8.272601606246393, −7.994092411514498, −6.911660596668879, −6.776226885018231, −6.149479579131446, −5.615437021962906, −4.770227276626618, −3.898248107063137, −3.362703978281655, −2.721675950131921, −2.302832780028075, −1.205039199763330, −0.6329586688441112,
0.6329586688441112, 1.205039199763330, 2.302832780028075, 2.721675950131921, 3.362703978281655, 3.898248107063137, 4.770227276626618, 5.615437021962906, 6.149479579131446, 6.776226885018231, 6.911660596668879, 7.994092411514498, 8.272601606246393, 8.785432962502342, 9.560833308513423, 9.825181723872391, 10.22174525065697, 10.60908313975156, 11.58689596411431, 12.00644742699426, 12.59895758600131, 13.13940929925929, 13.38783351927800, 14.20239856527712, 14.54143168122536
