L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s − 4·11-s + 12-s + 15-s + 16-s − 2·17-s − 18-s − 4·19-s + 20-s + 4·22-s − 8·23-s − 24-s + 25-s + 27-s + 6·29-s − 30-s − 8·31-s − 32-s − 4·33-s + 2·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s + 0.288·12-s + 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.852·22-s − 1.66·23-s − 0.204·24-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.182·30-s − 1.43·31-s − 0.176·32-s − 0.696·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.34543487371625, −13.04014036371546, −12.33009616395802, −11.96280999602699, −11.41537360138777, −10.74213460926137, −10.46855426567880, −10.01496877057143, −9.749922754126261, −8.987399620974114, −8.659661567546917, −8.282838681290569, −7.694052792708907, −7.475665533562253, −6.560410150083122, −6.468038536827033, −5.733313039151271, −5.226902043702164, −4.638215256636915, −4.029665725197424, −3.456855049432381, −2.695269600220366, −2.410750621020235, −1.802920020024841, −1.311128479279130, 0, 0,
1.311128479279130, 1.802920020024841, 2.410750621020235, 2.695269600220366, 3.456855049432381, 4.029665725197424, 4.638215256636915, 5.226902043702164, 5.733313039151271, 6.468038536827033, 6.560410150083122, 7.475665533562253, 7.694052792708907, 8.282838681290569, 8.659661567546917, 8.987399620974114, 9.749922754126261, 10.01496877057143, 10.46855426567880, 10.74213460926137, 11.41537360138777, 11.96280999602699, 12.33009616395802, 13.04014036371546, 13.34543487371625