| L(s) = 1 | + 1.24·2-s − 0.210·3-s − 6.44·4-s + 8.51·5-s − 0.262·6-s − 18.0·8-s − 26.9·9-s + 10.6·10-s + 4.14·11-s + 1.35·12-s + 18.8·13-s − 1.79·15-s + 29.1·16-s − 21.8·17-s − 33.5·18-s + 46.4·19-s − 54.8·20-s + 5.16·22-s + 82.4·23-s + 3.78·24-s − 52.5·25-s + 23.5·26-s + 11.3·27-s − 200.·29-s − 2.23·30-s + 276.·31-s + 180.·32-s + ⋯ |
| L(s) = 1 | + 0.440·2-s − 0.0404·3-s − 0.805·4-s + 0.761·5-s − 0.0178·6-s − 0.795·8-s − 0.998·9-s + 0.335·10-s + 0.113·11-s + 0.0326·12-s + 0.402·13-s − 0.0308·15-s + 0.455·16-s − 0.312·17-s − 0.439·18-s + 0.561·19-s − 0.613·20-s + 0.0500·22-s + 0.747·23-s + 0.0322·24-s − 0.420·25-s + 0.177·26-s + 0.0809·27-s − 1.28·29-s − 0.0135·30-s + 1.60·31-s + 0.996·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 1.24T + 8T^{2} \) |
| 3 | \( 1 + 0.210T + 27T^{2} \) |
| 5 | \( 1 - 8.51T + 125T^{2} \) |
| 11 | \( 1 - 4.14T + 1.33e3T^{2} \) |
| 13 | \( 1 - 18.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 21.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 46.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 82.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 200.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 276.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 217.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 429.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 208.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 415.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 176.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 166.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 325.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 809.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 201.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 627.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 69.4T + 4.93e5T^{2} \) |
| 83 | \( 1 + 707.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 862.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.05e3T + 9.12e5T^{2} \) |
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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
−8.424937257888558883806634161575, −7.49721752332139363943871991364, −6.33932405316156715347920443821, −5.77300440316726770332557995184, −5.18376733093133702936039531883, −4.25667895230646429101997113184, −3.32383483095653660722998054217, −2.50075781206129436123797759608, −1.17422366414697654159369710192, 0,
1.17422366414697654159369710192, 2.50075781206129436123797759608, 3.32383483095653660722998054217, 4.25667895230646429101997113184, 5.18376733093133702936039531883, 5.77300440316726770332557995184, 6.33932405316156715347920443821, 7.49721752332139363943871991364, 8.424937257888558883806634161575
