| L(s) = 1 | − 9.35·5-s − 8.35·7-s + 29.0·11-s + 36.7·13-s + 28.4·17-s − 73.7·19-s − 30.5·23-s − 37.4·25-s − 184.·29-s + 190.·31-s + 78.2·35-s + 160.·37-s − 209.·41-s − 116.·43-s − 281.·47-s − 273.·49-s − 397.·53-s − 272.·55-s + 378.·59-s − 297.·61-s − 344.·65-s + 827.·67-s + 729.·71-s + 1.10e3·73-s − 242.·77-s + 1.06e3·79-s + 590.·83-s + ⋯ |
| L(s) = 1 | − 0.836·5-s − 0.451·7-s + 0.796·11-s + 0.784·13-s + 0.405·17-s − 0.890·19-s − 0.276·23-s − 0.299·25-s − 1.18·29-s + 1.10·31-s + 0.377·35-s + 0.713·37-s − 0.798·41-s − 0.414·43-s − 0.873·47-s − 0.796·49-s − 1.03·53-s − 0.667·55-s + 0.834·59-s − 0.624·61-s − 0.656·65-s + 1.50·67-s + 1.21·71-s + 1.77·73-s − 0.359·77-s + 1.51·79-s + 0.780·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.469839414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.469839414\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 9.35T + 125T^{2} \) |
| 7 | \( 1 + 8.35T + 343T^{2} \) |
| 11 | \( 1 - 29.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 36.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 28.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 73.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 30.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 184.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 190.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 160.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 209.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 116.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 281.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 397.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 378.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 297.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 827.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 729.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.10e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.06e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 590.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 227.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.28e3T + 9.12e5T^{2} \) |
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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
−9.317816590807052843336545492320, −8.318800915735394409790268293055, −7.84999297854889725790760364258, −6.65027191037327028262207768562, −6.21536485216236749903271858315, −4.95296031590613117317430747382, −3.89372656435590185404461643122, −3.41567102481517557139408833453, −1.92838023100971863350204187665, −0.60916267696053042714447830025,
0.60916267696053042714447830025, 1.92838023100971863350204187665, 3.41567102481517557139408833453, 3.89372656435590185404461643122, 4.95296031590613117317430747382, 6.21536485216236749903271858315, 6.65027191037327028262207768562, 7.84999297854889725790760364258, 8.318800915735394409790268293055, 9.317816590807052843336545492320
