| L(s) = 1 | − 0.248·2-s − 7.93·4-s + 12.4·5-s + 3.95·8-s − 3.08·10-s − 60.3·11-s + 36.4·13-s + 62.5·16-s + 48.7·17-s − 50.5·19-s − 98.7·20-s + 14.9·22-s − 138.·23-s + 29.6·25-s − 9.03·26-s + 61.1·29-s − 1.16·31-s − 47.1·32-s − 12.0·34-s + 69.5·37-s + 12.5·38-s + 49.1·40-s − 308.·41-s + 174.·43-s + 478.·44-s + 34.4·46-s − 389.·47-s + ⋯ |
| L(s) = 1 | − 0.0877·2-s − 0.992·4-s + 1.11·5-s + 0.174·8-s − 0.0975·10-s − 1.65·11-s + 0.777·13-s + 0.976·16-s + 0.695·17-s − 0.610·19-s − 1.10·20-s + 0.144·22-s − 1.25·23-s + 0.236·25-s − 0.0681·26-s + 0.391·29-s − 0.00677·31-s − 0.260·32-s − 0.0609·34-s + 0.308·37-s + 0.0535·38-s + 0.194·40-s − 1.17·41-s + 0.618·43-s + 1.64·44-s + 0.110·46-s − 1.20·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 0.248T + 8T^{2} \) |
| 5 | \( 1 - 12.4T + 125T^{2} \) |
| 11 | \( 1 + 60.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 36.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 48.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 50.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 138.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 61.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 1.16T + 2.97e4T^{2} \) |
| 37 | \( 1 - 69.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 308.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 174.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 389.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 314.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 844.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 338.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 971.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 98.4T + 3.57e5T^{2} \) |
| 73 | \( 1 - 710.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 486.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 605.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 218.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 782.T + 9.12e5T^{2} \) |
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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
−10.15328840944638458773017871456, −9.503430972691324840678557170541, −8.409214981297927113919544277268, −7.79957486622522184973917061394, −6.16661235251940828925483697734, −5.50989562642621202121009616004, −4.51847962085970255774024643158, −3.10327213086883755845245355050, −1.67656957342707491011439612642, 0,
1.67656957342707491011439612642, 3.10327213086883755845245355050, 4.51847962085970255774024643158, 5.50989562642621202121009616004, 6.16661235251940828925483697734, 7.79957486622522184973917061394, 8.409214981297927113919544277268, 9.503430972691324840678557170541, 10.15328840944638458773017871456
