| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 3·11-s + 12-s − 6·13-s − 15-s + 16-s + 3·17-s − 18-s + 6·19-s − 20-s − 3·22-s − 23-s − 24-s + 25-s + 6·26-s + 27-s + 3·29-s + 30-s − 32-s + 3·33-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 + 0.904·11-s + 0.288·12-s − 1.66·13-s − 0.258·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 1.37·19-s − 0.223·20-s − 0.639·22-s − 0.208·23-s − 0.204·24-s + 1/5·25-s + 1.17·26-s + 0.192·27-s + 0.557·29-s + 0.182·30-s − 0.176·32-s + 0.522·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−14.77517094484527, −14.21893158501398, −13.94855391671909, −13.07221781288403, −12.38725639148488, −12.18611347420812, −11.60539552725396, −11.17278511019923, −10.37984096135527, −9.805411591289028, −9.552896824088995, −9.129772213615202, −8.345665036434130, −7.833351393345454, −7.559585511455676, −6.874409773064345, −6.530128946563272, −5.511775418886331, −5.094749646297035, −4.261871022603437, −3.722784680442806, −2.903631517043632, −2.599753168710671, −1.574754205342636, −1.014516831200962, 0,
1.014516831200962, 1.574754205342636, 2.599753168710671, 2.903631517043632, 3.722784680442806, 4.261871022603437, 5.094749646297035, 5.511775418886331, 6.530128946563272, 6.874409773064345, 7.559585511455676, 7.833351393345454, 8.345665036434130, 9.129772213615202, 9.552896824088995, 9.805411591289028, 10.37984096135527, 11.17278511019923, 11.60539552725396, 12.18611347420812, 12.38725639148488, 13.07221781288403, 13.94855391671909, 14.21893158501398, 14.77517094484527
