| L(s) = 1 | − 197. i·2-s − 2.27e4·4-s + 6.71e4i·5-s + 4.53e5·7-s + 1.24e6i·8-s + 1.32e7·10-s + 2.02e6i·11-s + 6.90e7·13-s − 8.97e7i·14-s − 1.25e8·16-s − 6.66e8i·17-s − 1.19e9·19-s − 1.52e9i·20-s + 4.00e8·22-s − 2.10e9i·23-s + ⋯ |
| L(s) = 1 | − 1.54i·2-s − 1.38·4-s + 0.859i·5-s + 0.551·7-s + 0.595i·8-s + 1.32·10-s + 0.103i·11-s + 1.10·13-s − 0.851i·14-s − 0.465·16-s − 1.62i·17-s − 1.34·19-s − 1.19i·20-s + 0.160·22-s − 0.617i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(1.569345316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.569345316\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 197. iT - 1.63e4T^{2} \) |
| 5 | \( 1 - 6.71e4iT - 6.10e9T^{2} \) |
| 7 | \( 1 - 4.53e5T + 6.78e11T^{2} \) |
| 11 | \( 1 - 2.02e6iT - 3.79e14T^{2} \) |
| 13 | \( 1 - 6.90e7T + 3.93e15T^{2} \) |
| 17 | \( 1 + 6.66e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 + 1.19e9T + 7.99e17T^{2} \) |
| 23 | \( 1 + 2.10e9iT - 1.15e19T^{2} \) |
| 29 | \( 1 + 2.11e10iT - 2.97e20T^{2} \) |
| 31 | \( 1 - 3.86e10T + 7.56e20T^{2} \) |
| 37 | \( 1 + 1.72e11T + 9.01e21T^{2} \) |
| 41 | \( 1 - 3.23e10iT - 3.79e22T^{2} \) |
| 43 | \( 1 - 5.94e10T + 7.38e22T^{2} \) |
| 47 | \( 1 - 1.00e11iT - 2.56e23T^{2} \) |
| 53 | \( 1 + 2.31e12iT - 1.37e24T^{2} \) |
| 59 | \( 1 - 2.97e11iT - 6.19e24T^{2} \) |
| 61 | \( 1 - 1.74e12T + 9.87e24T^{2} \) |
| 67 | \( 1 - 3.39e12T + 3.67e25T^{2} \) |
| 71 | \( 1 + 1.02e13iT - 8.27e25T^{2} \) |
| 73 | \( 1 + 5.19e12T + 1.22e26T^{2} \) |
| 79 | \( 1 + 1.50e13T + 3.68e26T^{2} \) |
| 83 | \( 1 + 3.76e13iT - 7.36e26T^{2} \) |
| 89 | \( 1 + 6.45e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 + 1.57e13T + 6.52e27T^{2} \) |
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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
−13.26756171145188959484977820368, −11.82884502387886147434997401097, −11.01488908647714147273521030093, −10.07210494105815184907707716579, −8.564795962751385917745591393716, −6.66727627281360357325577099703, −4.54233262796765001584589712095, −3.13371934686053897206560121690, −2.01232974651216085067637829258, −0.49738787547196007130855852066,
1.41449003820266548618067860770, 4.17670027732078644464582615456, 5.44267854887156965292637676894, 6.59249427128767193145692946314, 8.278089891500644689839457637623, 8.710444942367302712121627358016, 10.80006707711576314720658861104, 12.61419733364363518226115713464, 13.78649741873386727936978906760, 14.97968094908074054051679898980
