| L(s) = 1 | − 3-s − 2·7-s + 9-s + 4·13-s − 17-s + 2·21-s − 8·23-s − 5·25-s − 27-s − 2·29-s − 8·31-s + 6·37-s − 4·39-s − 2·41-s − 8·43-s + 6·47-s − 3·49-s + 51-s + 2·53-s − 6·59-s − 2·61-s − 2·63-s + 8·69-s − 8·71-s + 2·73-s + 5·75-s + 2·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 1.10·13-s − 0.242·17-s + 0.436·21-s − 1.66·23-s − 25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.986·37-s − 0.640·39-s − 0.312·41-s − 1.21·43-s + 0.875·47-s − 3/7·49-s + 0.140·51-s + 0.274·53-s − 0.781·59-s − 0.256·61-s − 0.251·63-s + 0.963·69-s − 0.949·71-s + 0.234·73-s + 0.577·75-s + 0.225·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3003616189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3003616189\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−13.63318647437188, −13.30447342742587, −12.80645209560984, −12.27961495399728, −11.79087410977305, −11.35358130982105, −10.83192499729469, −10.36942525596308, −9.815412149798696, −9.408290371986083, −8.883272178657290, −8.192404908112754, −7.771329299799459, −7.175064693170119, −6.490896957043070, −6.145665752538377, −5.730502052513403, −5.177927825096170, −4.315319890110251, −3.848731550727189, −3.484590211470984, −2.600104119287988, −1.852043119152467, −1.301840193484850, −0.1797505050093142,
0.1797505050093142, 1.301840193484850, 1.852043119152467, 2.600104119287988, 3.484590211470984, 3.848731550727189, 4.315319890110251, 5.177927825096170, 5.730502052513403, 6.145665752538377, 6.490896957043070, 7.175064693170119, 7.771329299799459, 8.192404908112754, 8.883272178657290, 9.408290371986083, 9.815412149798696, 10.36942525596308, 10.83192499729469, 11.35358130982105, 11.79087410977305, 12.27961495399728, 12.80645209560984, 13.30447342742587, 13.63318647437188
