| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 2·7-s − 8-s − 2·9-s − 3·11-s + 12-s + 13-s − 2·14-s + 16-s − 8·17-s + 2·18-s + 2·21-s + 3·22-s − 4·23-s − 24-s − 26-s − 5·27-s + 2·28-s − 29-s − 3·31-s − 32-s − 3·33-s + 8·34-s − 2·36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s − 2/3·9-s − 0.904·11-s + 0.288·12-s + 0.277·13-s − 0.534·14-s + 1/4·16-s − 1.94·17-s + 0.471·18-s + 0.436·21-s + 0.639·22-s − 0.834·23-s − 0.204·24-s − 0.196·26-s − 0.962·27-s + 0.377·28-s − 0.185·29-s − 0.538·31-s − 0.176·32-s − 0.522·33-s + 1.37·34-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−8.926995527685451598137983295064, −8.389384753088059863163172626096, −7.81565944790560260026240611700, −6.89413781928069079756972828354, −5.89877989020180959127647474305, −4.96570329407044523914241607212, −3.82505096342084054880139925422, −2.58882766939603611448371560350, −1.89442567285374802225669499718, 0,
1.89442567285374802225669499718, 2.58882766939603611448371560350, 3.82505096342084054880139925422, 4.96570329407044523914241607212, 5.89877989020180959127647474305, 6.89413781928069079756972828354, 7.81565944790560260026240611700, 8.389384753088059863163172626096, 8.926995527685451598137983295064
