| L(s) = 1 | + 4.59·2-s − 3·3-s + 13.1·4-s + 5·5-s − 13.7·6-s + 20.6·7-s + 23.4·8-s + 9·9-s + 22.9·10-s + 11·11-s − 39.3·12-s − 15.6·13-s + 94.8·14-s − 15·15-s + 3.04·16-s + 72.9·17-s + 41.3·18-s + 61.0·19-s + 65.5·20-s − 61.9·21-s + 50.5·22-s − 13.6·23-s − 70.4·24-s + 25·25-s − 71.9·26-s − 27·27-s + 270.·28-s + ⋯ |
| L(s) = 1 | + 1.62·2-s − 0.577·3-s + 1.63·4-s + 0.447·5-s − 0.937·6-s + 1.11·7-s + 1.03·8-s + 0.333·9-s + 0.726·10-s + 0.301·11-s − 0.946·12-s − 0.334·13-s + 1.81·14-s − 0.258·15-s + 0.0475·16-s + 1.04·17-s + 0.541·18-s + 0.737·19-s + 0.733·20-s − 0.643·21-s + 0.489·22-s − 0.123·23-s − 0.599·24-s + 0.200·25-s − 0.542·26-s − 0.192·27-s + 1.82·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.030201939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.030201939\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 4.59T + 8T^{2} \) |
| 7 | \( 1 - 20.6T + 343T^{2} \) |
| 13 | \( 1 + 15.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 72.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 61.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 13.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 31.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 243.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 65.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 109.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 121.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 519.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 542.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 109.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 89.6T + 2.26e5T^{2} \) |
| 67 | \( 1 - 488.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 837.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 351.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 831.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.38e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.52e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 426.T + 9.12e5T^{2} \) |
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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
−12.39306740734905279527055484718, −11.66292431247261758265222590177, −10.88259697363829445524401388231, −9.549849553089363393120341412171, −7.83122693648406139570091848270, −6.63572952843573568707216839022, −5.44902853802903441730155701058, −4.91558040455083219916512788620, −3.50125993267377666422717976530, −1.74409730451185268088544075927,
1.74409730451185268088544075927, 3.50125993267377666422717976530, 4.91558040455083219916512788620, 5.44902853802903441730155701058, 6.63572952843573568707216839022, 7.83122693648406139570091848270, 9.549849553089363393120341412171, 10.88259697363829445524401388231, 11.66292431247261758265222590177, 12.39306740734905279527055484718
