| L(s) = 1 | + 3-s − 2.37·5-s + 2.37·7-s + 9-s − 6.37·11-s + 4·13-s − 2.37·15-s + 0.372·17-s − 19-s + 2.37·21-s + 6.74·23-s + 0.627·25-s + 27-s − 2.74·29-s + 6·31-s − 6.37·33-s − 5.62·35-s + 4·37-s + 4·39-s − 2.74·41-s + 1.62·43-s − 2.37·45-s − 9.11·47-s − 1.37·49-s + 0.372·51-s + 10·53-s + 15.1·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.06·5-s + 0.896·7-s + 0.333·9-s − 1.92·11-s + 1.10·13-s − 0.612·15-s + 0.0902·17-s − 0.229·19-s + 0.517·21-s + 1.40·23-s + 0.125·25-s + 0.192·27-s − 0.509·29-s + 1.07·31-s − 1.10·33-s − 0.951·35-s + 0.657·37-s + 0.640·39-s − 0.428·41-s + 0.248·43-s − 0.353·45-s − 1.32·47-s − 0.196·49-s + 0.0521·51-s + 1.37·53-s + 2.03·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.936536424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.936536424\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2.37T + 5T^{2} \) |
| 7 | \( 1 - 2.37T + 7T^{2} \) |
| 11 | \( 1 + 6.37T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 0.372T + 17T^{2} \) |
| 23 | \( 1 - 6.74T + 23T^{2} \) |
| 29 | \( 1 + 2.74T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 2.74T + 41T^{2} \) |
| 43 | \( 1 - 1.62T + 43T^{2} \) |
| 47 | \( 1 + 9.11T + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 - 8.74T + 59T^{2} \) |
| 61 | \( 1 + 3.62T + 61T^{2} \) |
| 67 | \( 1 - 4.74T + 67T^{2} \) |
| 71 | \( 1 - 3.25T + 71T^{2} \) |
| 73 | \( 1 - 8.37T + 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 18.7T + 97T^{2} \) |
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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
−8.315290778295756825548502918308, −7.927327993456255391767692750192, −7.41657195224643721874673776057, −6.41385930290174518044740914755, −5.26885393112931481066423253184, −4.76297351360025722761181220129, −3.81401234122966478464312576732, −3.05832415914518276509532417836, −2.12749494199241867888086155128, −0.793287669969702985952826054578,
0.793287669969702985952826054578, 2.12749494199241867888086155128, 3.05832415914518276509532417836, 3.81401234122966478464312576732, 4.76297351360025722761181220129, 5.26885393112931481066423253184, 6.41385930290174518044740914755, 7.41657195224643721874673776057, 7.927327993456255391767692750192, 8.315290778295756825548502918308
