| L(s) = 1 | + 8.54·5-s − 31.7i·11-s − 9.92i·13-s + 127.·17-s − 116. i·19-s − 64.4i·23-s − 52.0·25-s + 113. i·29-s − 7.31i·31-s + 369.·37-s − 211.·41-s − 432.·43-s − 400.·47-s − 140. i·53-s − 270. i·55-s + ⋯ |
| L(s) = 1 | + 0.764·5-s − 0.869i·11-s − 0.211i·13-s + 1.81·17-s − 1.40i·19-s − 0.584i·23-s − 0.416·25-s + 0.723i·29-s − 0.0423i·31-s + 1.64·37-s − 0.807·41-s − 1.53·43-s − 1.24·47-s − 0.364i·53-s − 0.664i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0980 + 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0980 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.218250976\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.218250976\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 8.54T + 125T^{2} \) |
| 11 | \( 1 + 31.7iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 9.92iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 127.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 116. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 64.4iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 113. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 7.31iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 369.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 211.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 432.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 400.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 140. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 518.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 27.2iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 136.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 604. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 48.4iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 831.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 37.2T + 5.71e5T^{2} \) |
| 89 | \( 1 - 470.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 522. iT - 9.12e5T^{2} \) |
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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
−8.675208660455559660929224066863, −8.066311099787182865618223612859, −7.07978674333798796809631558861, −6.23090046200177747607199516245, −5.52462861961035714508913241505, −4.79350932952331722486045707394, −3.46693398853759509238349696541, −2.76640011374673120965321630495, −1.51606060253736678874220893033, −0.47056289707358519096659551266,
1.25352471164592928184611347040, 2.00589701272487630450177505715, 3.22184071633850211350325191715, 4.13839853857809594739126102005, 5.26828625540414930701101775121, 5.83757583733492618204895308901, 6.69029294879706866747924750088, 7.72334713232588046537015120182, 8.161187500581832346913001732140, 9.440183437256241414690386870921
