| L(s) = 1 | + 0.519·5-s − 10.4·7-s + 14.1·11-s + 5.39i·13-s − 24.9i·17-s − 13.3i·19-s + 41.1i·23-s − 24.7·25-s + 7.85·29-s + 26.1·31-s − 5.40·35-s + 1.53i·37-s − 21.3i·41-s + 64.1i·43-s − 19.8i·47-s + ⋯ |
| L(s) = 1 | + 0.103·5-s − 1.48·7-s + 1.28·11-s + 0.414i·13-s − 1.46i·17-s − 0.701i·19-s + 1.78i·23-s − 0.989·25-s + 0.270·29-s + 0.844·31-s − 0.154·35-s + 0.0415i·37-s − 0.520i·41-s + 1.49i·43-s − 0.422i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1738989269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1738989269\) |
| \(L(2)\) |
|
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 | \( 1 - 0.519T + 25T^{2} \) |
| 7 | \( 1 + 10.4T + 49T^{2} \) |
| 11 | \( 1 - 14.1T + 121T^{2} \) |
| 13 | \( 1 - 5.39iT - 169T^{2} \) |
| 17 | \( 1 + 24.9iT - 289T^{2} \) |
| 19 | \( 1 + 13.3iT - 361T^{2} \) |
| 23 | \( 1 - 41.1iT - 529T^{2} \) |
| 29 | \( 1 - 7.85T + 841T^{2} \) |
| 31 | \( 1 - 26.1T + 961T^{2} \) |
| 37 | \( 1 - 1.53iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 21.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 64.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 19.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 68.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 67.8T + 3.48e3T^{2} \) |
| 61 | \( 1 + 58.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 56.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 107. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 7.73T + 5.32e3T^{2} \) |
| 79 | \( 1 + 64.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + 125.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 59.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 138.T + 9.40e3T^{2} \) |
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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
−9.569377535623856274766545123328, −9.061114750796733921586653119452, −7.81561055333648892425481602526, −6.89557300040847094125086028468, −6.47950700120806589740595948386, −5.56516917372846762978971451725, −4.45094196809876111619545315339, −3.51813412380126603590972509976, −2.75771016113854589083801135459, −1.31925244643897940139894191315,
0.04747946201853915740963549078, 1.46384406458317475084232638393, 2.78437198228318529970451670666, 3.72344993174596777615413403219, 4.37165956298899778802304198396, 5.94633336394884657187384834548, 6.22409009029856025281243888617, 6.96840476987480457809936078040, 8.140731383246826991739191798170, 8.777254474017677873917204652961
