L(s) = 1 | − 2-s − 3-s − 4-s + 2·5-s + 6-s + 7-s + 3·8-s + 9-s − 2·10-s − 4·11-s + 12-s − 2·13-s − 14-s − 2·15-s − 16-s − 18-s + 4·19-s − 2·20-s − 21-s + 4·22-s − 3·24-s − 25-s + 2·26-s − 27-s − 28-s + 2·29-s + 2·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s + 0.288·12-s − 0.554·13-s − 0.267·14-s − 0.516·15-s − 1/4·16-s − 0.235·18-s + 0.917·19-s − 0.447·20-s − 0.218·21-s + 0.852·22-s − 0.612·24-s − 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.188·28-s + 0.371·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8752924544\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8752924544\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−8.150173736744291796300091652744, −7.43276089750609294899404044756, −6.83652065925235047357009366271, −5.70635600713953756001504269408, −5.24463571202663205176745275817, −4.77630342481941975935989178304, −3.69050895952500294993723894736, −2.48232080233279175904585277989, −1.65235155148622815444498272942, −0.57807668787933325619389723813,
0.57807668787933325619389723813, 1.65235155148622815444498272942, 2.48232080233279175904585277989, 3.69050895952500294993723894736, 4.77630342481941975935989178304, 5.24463571202663205176745275817, 5.70635600713953756001504269408, 6.83652065925235047357009366271, 7.43276089750609294899404044756, 8.150173736744291796300091652744