| L(s) = 1 | − 2.41·2-s − 3-s + 3.82·4-s − 5-s + 2.41·6-s + 0.828·7-s − 4.41·8-s + 9-s + 2.41·10-s − 11-s − 3.82·12-s − 5.65·13-s − 1.99·14-s + 15-s + 2.99·16-s − 1.17·17-s − 2.41·18-s − 6.82·19-s − 3.82·20-s − 0.828·21-s + 2.41·22-s − 4·23-s + 4.41·24-s + 25-s + 13.6·26-s − 27-s + 3.17·28-s + ⋯ |
| L(s) = 1 | − 1.70·2-s − 0.577·3-s + 1.91·4-s − 0.447·5-s + 0.985·6-s + 0.313·7-s − 1.56·8-s + 0.333·9-s + 0.763·10-s − 0.301·11-s − 1.10·12-s − 1.56·13-s − 0.534·14-s + 0.258·15-s + 0.749·16-s − 0.284·17-s − 0.569·18-s − 1.56·19-s − 0.856·20-s − 0.180·21-s + 0.514·22-s − 0.834·23-s + 0.901·24-s + 0.200·25-s + 2.67·26-s − 0.192·27-s + 0.599·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 7 | \( 1 - 0.828T + 7T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 + 1.17T + 17T^{2} \) |
| 19 | \( 1 + 6.82T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 - 4.82T + 41T^{2} \) |
| 43 | \( 1 + 8.82T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 9.31T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 - 2.34T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 + 10T + 83T^{2} \) |
| 89 | \( 1 - 3.65T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−11.89236308217832953012385481959, −11.08616747244285173223687672393, −10.24270735616098881995544842593, −9.366705082037274225159796771471, −8.145492917234143663137211812727, −7.44551346438669193859360856384, −6.31227099398064812462775450942, −4.60397357310367788518780208858, −2.19032666796141313339072907284, 0,
2.19032666796141313339072907284, 4.60397357310367788518780208858, 6.31227099398064812462775450942, 7.44551346438669193859360856384, 8.145492917234143663137211812727, 9.366705082037274225159796771471, 10.24270735616098881995544842593, 11.08616747244285173223687672393, 11.89236308217832953012385481959
