| L(s) = 1 | + 2.79e3i·3-s + 1.07e5·5-s − 1.45e6i·7-s − 3.03e6·9-s − 2.07e7i·11-s + 5.92e7·13-s + 3.01e8i·15-s − 4.48e7·17-s − 8.57e8i·19-s + 4.08e9·21-s − 1.37e9i·23-s + 5.52e9·25-s + 4.87e9i·27-s + 3.05e10·29-s + 3.46e10i·31-s + ⋯ |
| L(s) = 1 | + 1.27i·3-s + 1.38·5-s − 1.77i·7-s − 0.635·9-s − 1.06i·11-s + 0.944·13-s + 1.76i·15-s − 0.109·17-s − 0.959i·19-s + 2.26·21-s − 0.402i·23-s + 0.905·25-s + 0.466i·27-s + 1.76·29-s + 1.26i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(2.48669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.48669\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 2.79e3iT - 4.78e6T^{2} \) |
| 5 | \( 1 - 1.07e5T + 6.10e9T^{2} \) |
| 7 | \( 1 + 1.45e6iT - 6.78e11T^{2} \) |
| 11 | \( 1 + 2.07e7iT - 3.79e14T^{2} \) |
| 13 | \( 1 - 5.92e7T + 3.93e15T^{2} \) |
| 17 | \( 1 + 4.48e7T + 1.68e17T^{2} \) |
| 19 | \( 1 + 8.57e8iT - 7.99e17T^{2} \) |
| 23 | \( 1 + 1.37e9iT - 1.15e19T^{2} \) |
| 29 | \( 1 - 3.05e10T + 2.97e20T^{2} \) |
| 31 | \( 1 - 3.46e10iT - 7.56e20T^{2} \) |
| 37 | \( 1 + 1.25e11T + 9.01e21T^{2} \) |
| 41 | \( 1 + 4.14e10T + 3.79e22T^{2} \) |
| 43 | \( 1 - 1.00e11iT - 7.38e22T^{2} \) |
| 47 | \( 1 - 4.42e9iT - 2.56e23T^{2} \) |
| 53 | \( 1 - 4.73e11T + 1.37e24T^{2} \) |
| 59 | \( 1 - 3.16e12iT - 6.19e24T^{2} \) |
| 61 | \( 1 + 5.85e10T + 9.87e24T^{2} \) |
| 67 | \( 1 + 4.17e12iT - 3.67e25T^{2} \) |
| 71 | \( 1 + 1.56e13iT - 8.27e25T^{2} \) |
| 73 | \( 1 - 1.46e13T + 1.22e26T^{2} \) |
| 79 | \( 1 - 8.37e12iT - 3.68e26T^{2} \) |
| 83 | \( 1 + 2.46e13iT - 7.36e26T^{2} \) |
| 89 | \( 1 - 3.58e12T + 1.95e27T^{2} \) |
| 97 | \( 1 - 6.51e13T + 6.52e27T^{2} \) |
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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
−15.92163319199249814649902996344, −14.05635576943637861723533342956, −13.50530981485950887714662155355, −10.76743293064391824892197105426, −10.28298471431699190060176507117, −8.860837924954879344064903426070, −6.52727457433197210817288100159, −4.83610788376593993730034581490, −3.39969698437429782264561574663, −1.01116116653815490583030109822,
1.58261540363514476121516155053, 2.33982814864191627718871810810, 5.57922377786305722307423218722, 6.53256606930846835819564950845, 8.423837012272146087866772988024, 9.809904113805586099616649339723, 11.99911416829980436921331354013, 12.86967146733674543722078485816, 14.02853446449504548523313315972, 15.53508638071805642537672524827
