| L(s) = 1 | + i·2-s − 4-s − 2.37i·7-s − i·8-s − 1.37·11-s − 4.74i·13-s + 2.37·14-s + 16-s + 7.37i·17-s − 3.37·19-s − 1.37i·22-s − 4.37i·23-s + 4.74·26-s + 2.37i·28-s − 4.37·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.896i·7-s − 0.353i·8-s − 0.413·11-s − 1.31i·13-s + 0.634·14-s + 0.250·16-s + 1.78i·17-s − 0.773·19-s − 0.292i·22-s − 0.911i·23-s + 0.930·26-s + 0.448i·28-s − 0.811·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6086021668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6086021668\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2.37iT - 7T^{2} \) |
| 11 | \( 1 + 1.37T + 11T^{2} \) |
| 13 | \( 1 + 4.74iT - 13T^{2} \) |
| 17 | \( 1 - 7.37iT - 17T^{2} \) |
| 19 | \( 1 + 3.37T + 19T^{2} \) |
| 23 | \( 1 + 4.37iT - 23T^{2} \) |
| 29 | \( 1 + 4.37T + 29T^{2} \) |
| 31 | \( 1 + 6.74T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - 11.3iT - 43T^{2} \) |
| 47 | \( 1 - 1.62iT - 47T^{2} \) |
| 53 | \( 1 - 11.4iT - 53T^{2} \) |
| 59 | \( 1 + 1.37T + 59T^{2} \) |
| 61 | \( 1 - 9.11T + 61T^{2} \) |
| 67 | \( 1 + 7iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 14.1iT - 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 - 1.62iT - 83T^{2} \) |
| 89 | \( 1 + 1.11T + 89T^{2} \) |
| 97 | \( 1 + 2.62iT - 97T^{2} \) |
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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
−8.521678815039712742900869830554, −7.87428527804339569401342023064, −7.47520043967607982921676775156, −6.46218432317787708662318016181, −5.92338202898374646995267932321, −5.14353693939201211405620119315, −4.15176120743745036296161531604, −3.66263144999502433870780034980, −2.42725216253301458428984407423, −1.07878233351455661900915497336,
0.18794430929054974404636140227, 1.81820227111681070493436301637, 2.37553822442820959394192063021, 3.37569271593588456094873145576, 4.22979064219176622702233612220, 5.17583377473446632839253340236, 5.60508446893743978352863141869, 6.78087182426535111887236104297, 7.34634111527031899732735256990, 8.356990197076206731038520457858
