| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 2·7-s − 8-s + 9-s + 10-s − 2·11-s + 12-s + 13-s − 2·14-s − 15-s + 16-s + 2·17-s − 18-s + 6·19-s − 20-s + 2·21-s + 2·22-s − 2·23-s − 24-s + 25-s − 26-s + 27-s + 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s + 0.277·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 1.37·19-s − 0.223·20-s + 0.436·21-s + 0.426·22-s − 0.417·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−12.77117732672810, −12.27084460926433, −11.84232136303131, −11.43961452855406, −10.97815102709077, −10.52972058969347, −10.00759617375798, −9.688550600194321, −9.076771053470867, −8.658042553315397, −8.220934557029555, −7.821262730713152, −7.409864533360804, −7.141628114209042, −6.417811349994533, −5.746640771484761, −5.344501145843146, −4.821358790807255, −4.194038022279773, −3.581557168977493, −3.164377855857212, −2.585867568983225, −1.967228137462458, −1.374064749234271, −0.8592595433475485, 0,
0.8592595433475485, 1.374064749234271, 1.967228137462458, 2.585867568983225, 3.164377855857212, 3.581557168977493, 4.194038022279773, 4.821358790807255, 5.344501145843146, 5.746640771484761, 6.417811349994533, 7.141628114209042, 7.409864533360804, 7.821262730713152, 8.220934557029555, 8.658042553315397, 9.076771053470867, 9.688550600194321, 10.00759617375798, 10.52972058969347, 10.97815102709077, 11.43961452855406, 11.84232136303131, 12.27084460926433, 12.77117732672810
