| L(s) = 1 | − 3-s − 2·7-s + 9-s − 11-s − 3·13-s + 17-s − 6·19-s + 2·21-s + 6·23-s − 27-s + 9·29-s + 4·31-s + 33-s + 6·37-s + 3·39-s − 2·43-s − 13·47-s − 3·49-s − 51-s + 3·53-s + 6·57-s − 14·59-s − 12·61-s − 2·63-s − 7·67-s − 6·69-s − 6·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.832·13-s + 0.242·17-s − 1.37·19-s + 0.436·21-s + 1.25·23-s − 0.192·27-s + 1.67·29-s + 0.718·31-s + 0.174·33-s + 0.986·37-s + 0.480·39-s − 0.304·43-s − 1.89·47-s − 3/7·49-s − 0.140·51-s + 0.412·53-s + 0.794·57-s − 1.82·59-s − 1.53·61-s − 0.251·63-s − 0.855·67-s − 0.722·69-s − 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4257717071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4257717071\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−12.90234902404773, −12.51522630250797, −11.94715326230048, −11.77957487032056, −10.94186731062807, −10.64211902457649, −10.19953032745262, −9.802474529239779, −9.219501977765639, −8.840598170405135, −8.128177838458601, −7.770978107744248, −7.158478588311456, −6.580229143989237, −6.300696579645239, −5.931913164455471, −4.974112945138797, −4.778656965514587, −4.379586755751734, −3.513176830833450, −2.837037356310042, −2.696286598927756, −1.700545759767920, −1.094260629072686, −0.2025094934287384,
0.2025094934287384, 1.094260629072686, 1.700545759767920, 2.696286598927756, 2.837037356310042, 3.513176830833450, 4.379586755751734, 4.778656965514587, 4.974112945138797, 5.931913164455471, 6.300696579645239, 6.580229143989237, 7.158478588311456, 7.770978107744248, 8.128177838458601, 8.840598170405135, 9.219501977765639, 9.802474529239779, 10.19953032745262, 10.64211902457649, 10.94186731062807, 11.77957487032056, 11.94715326230048, 12.51522630250797, 12.90234902404773
