| L(s) = 1 | + 1.41i·2-s − 2.00·4-s − 5·7-s − 2.82i·8-s − 1.41i·11-s + 9·13-s − 7.07i·14-s + 4.00·16-s − 11.3i·17-s − 21·19-s + 2.00·22-s + 1.41i·23-s + 12.7i·26-s + 10.0·28-s + 38.1i·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s − 0.714·7-s − 0.353i·8-s − 0.128i·11-s + 0.692·13-s − 0.505i·14-s + 0.250·16-s − 0.665i·17-s − 1.10·19-s + 0.0909·22-s + 0.0614i·23-s + 0.489i·26-s + 0.357·28-s + 1.31i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.429893468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.429893468\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 5T + 49T^{2} \) |
| 11 | \( 1 + 1.41iT - 121T^{2} \) |
| 13 | \( 1 - 9T + 169T^{2} \) |
| 17 | \( 1 + 11.3iT - 289T^{2} \) |
| 19 | \( 1 + 21T + 361T^{2} \) |
| 23 | \( 1 - 1.41iT - 529T^{2} \) |
| 29 | \( 1 - 38.1iT - 841T^{2} \) |
| 31 | \( 1 - 40T + 961T^{2} \) |
| 37 | \( 1 - 25T + 1.36e3T^{2} \) |
| 41 | \( 1 + 52.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 64T + 1.84e3T^{2} \) |
| 47 | \( 1 + 22.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 72.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 90.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 97T + 3.72e3T^{2} \) |
| 67 | \( 1 - 131T + 4.48e3T^{2} \) |
| 71 | \( 1 - 89.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 17T + 5.32e3T^{2} \) |
| 79 | \( 1 - 117T + 6.24e3T^{2} \) |
| 83 | \( 1 + 57.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 147. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 41T + 9.40e3T^{2} \) |
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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
−9.347961007809038736370674560542, −8.843336536976711612009453967534, −7.976143456787218784838568401700, −7.05338880430828772794540463433, −6.37362001659865149908092510101, −5.65713155134435834754461036828, −4.58525772092427123404811728486, −3.68754695833858244509889452591, −2.58809173768703057628543248798, −0.886912598145638634744487193349,
0.54737041543364582378672122098, 1.93203843769617748910126047035, 2.98103994201911336432867238980, 3.94980913037106837859697732271, 4.70466518296137287597086681587, 6.11669022322999853965129551991, 6.42977018471664455354639522603, 7.88768964287094247262380070575, 8.435046346267932872607269442227, 9.466545068184599921199674824625
