| L(s) = 1 | + 13.5·5-s − 27.6·7-s − 13.4·11-s + 54.6·13-s − 99.5·17-s + 2.94·19-s + 140.·23-s + 58.5·25-s + 21.8·29-s + 185.·31-s − 374.·35-s + 24.0·37-s − 441.·41-s − 207.·43-s − 153.·47-s + 421.·49-s − 162·53-s − 182.·55-s − 245.·59-s − 416.·61-s + 740.·65-s − 695.·67-s + 493.·71-s + 220.·73-s + 371.·77-s − 711.·79-s + 204.·83-s + ⋯ |
| L(s) = 1 | + 1.21·5-s − 1.49·7-s − 0.368·11-s + 1.16·13-s − 1.42·17-s + 0.0356·19-s + 1.26·23-s + 0.468·25-s + 0.139·29-s + 1.07·31-s − 1.80·35-s + 0.106·37-s − 1.68·41-s − 0.736·43-s − 0.475·47-s + 1.22·49-s − 0.419·53-s − 0.446·55-s − 0.541·59-s − 0.875·61-s + 1.41·65-s − 1.26·67-s + 0.824·71-s + 0.353·73-s + 0.550·77-s − 1.01·79-s + 0.270·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 13.5T + 125T^{2} \) |
| 7 | \( 1 + 27.6T + 343T^{2} \) |
| 11 | \( 1 + 13.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 54.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 99.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 2.94T + 6.85e3T^{2} \) |
| 23 | \( 1 - 140.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 21.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 185.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 24.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 441.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 207.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 153.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 162T + 1.48e5T^{2} \) |
| 59 | \( 1 + 245.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 416.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 695.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 493.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 220.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 711.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 204.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 428.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 209.T + 9.12e5T^{2} \) |
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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
−9.068318720372667549832167453729, −8.281964085619308525111270201282, −6.77127341908595205078724527787, −6.51531758019142584700155530835, −5.70692653374876391050399044908, −4.67932082916092232648062744936, −3.39956645596540816058005038332, −2.63620227580743218370354476484, −1.43467498436652984397724960470, 0,
1.43467498436652984397724960470, 2.63620227580743218370354476484, 3.39956645596540816058005038332, 4.67932082916092232648062744936, 5.70692653374876391050399044908, 6.51531758019142584700155530835, 6.77127341908595205078724527787, 8.281964085619308525111270201282, 9.068318720372667549832167453729
