| L(s) = 1 | + 2-s − 2.41·3-s + 4-s − 2.64·5-s − 2.41·6-s + 8-s + 2.84·9-s − 2.64·10-s + 5.04·11-s − 2.41·12-s + 4.59·13-s + 6.39·15-s + 16-s − 4.48·17-s + 2.84·18-s − 1.98·19-s − 2.64·20-s + 5.04·22-s + 3.40·23-s − 2.41·24-s + 2.00·25-s + 4.59·26-s + 0.383·27-s − 5.50·29-s + 6.39·30-s − 0.322·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.39·3-s + 0.5·4-s − 1.18·5-s − 0.986·6-s + 0.353·8-s + 0.947·9-s − 0.836·10-s + 1.52·11-s − 0.697·12-s + 1.27·13-s + 1.65·15-s + 0.250·16-s − 1.08·17-s + 0.669·18-s − 0.455·19-s − 0.591·20-s + 1.07·22-s + 0.709·23-s − 0.493·24-s + 0.400·25-s + 0.901·26-s + 0.0738·27-s − 1.02·29-s + 1.16·30-s − 0.0580·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.389385511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.389385511\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 5 | \( 1 + 2.64T + 5T^{2} \) |
| 11 | \( 1 - 5.04T + 11T^{2} \) |
| 13 | \( 1 - 4.59T + 13T^{2} \) |
| 17 | \( 1 + 4.48T + 17T^{2} \) |
| 19 | \( 1 + 1.98T + 19T^{2} \) |
| 23 | \( 1 - 3.40T + 23T^{2} \) |
| 29 | \( 1 + 5.50T + 29T^{2} \) |
| 31 | \( 1 + 0.322T + 31T^{2} \) |
| 37 | \( 1 + 8.69T + 37T^{2} \) |
| 43 | \( 1 + 8.64T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 - 2.26T + 53T^{2} \) |
| 59 | \( 1 - 5.74T + 59T^{2} \) |
| 61 | \( 1 - 9.18T + 61T^{2} \) |
| 67 | \( 1 - 2.97T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 2.17T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 + 0.927T + 83T^{2} \) |
| 89 | \( 1 + 5.92T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−8.563138740375746398273342744629, −7.29530120966674069263714976360, −6.78046816982668358511063009222, −6.24484723384621241156766315866, −5.48584688198261013085263267304, −4.61261913926701350174651701951, −3.95314130568722677289102845498, −3.47272635123412211907429258396, −1.79879160289195411612233002032, −0.66561555217744342627535627541,
0.66561555217744342627535627541, 1.79879160289195411612233002032, 3.47272635123412211907429258396, 3.95314130568722677289102845498, 4.61261913926701350174651701951, 5.48584688198261013085263267304, 6.24484723384621241156766315866, 6.78046816982668358511063009222, 7.29530120966674069263714976360, 8.563138740375746398273342744629
