| L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s + 2·7-s − 8-s + 9-s + 2·10-s − 4·11-s − 12-s − 4·13-s − 2·14-s + 2·15-s + 16-s − 2·17-s − 18-s − 8·19-s − 2·20-s − 2·21-s + 4·22-s + 4·23-s + 24-s − 25-s + 4·26-s − 27-s + 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.20·11-s − 0.288·12-s − 1.10·13-s − 0.534·14-s + 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 1.83·19-s − 0.447·20-s − 0.436·21-s + 0.852·22-s + 0.834·23-s + 0.204·24-s − 1/5·25-s + 0.784·26-s − 0.192·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 246 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 246 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−11.29376271550970759508344963509, −10.86877357543799825273108323136, −9.810852451692795367975020441244, −8.467746331732822704290604152799, −7.75573819994472569207185165512, −6.85698182335088525790063480467, −5.36274716754459095896224840036, −4.29114537424233854905773976055, −2.32002779007866751340365487441, 0,
2.32002779007866751340365487441, 4.29114537424233854905773976055, 5.36274716754459095896224840036, 6.85698182335088525790063480467, 7.75573819994472569207185165512, 8.467746331732822704290604152799, 9.810852451692795367975020441244, 10.86877357543799825273108323136, 11.29376271550970759508344963509
