| L(s) = 1 | + 365. i·3-s − 321.·5-s − 7.74e3i·7-s − 7.47e4·9-s + 6.30e4i·11-s + 5.88e5·13-s − 1.17e5i·15-s − 1.10e6·17-s − 1.40e6i·19-s + 2.83e6·21-s + 8.10e6i·23-s − 9.66e6·25-s − 5.72e6i·27-s − 7.53e6·29-s − 3.74e7i·31-s + ⋯ |
| L(s) = 1 | + 1.50i·3-s − 0.102·5-s − 0.460i·7-s − 1.26·9-s + 0.391i·11-s + 1.58·13-s − 0.154i·15-s − 0.778·17-s − 0.568i·19-s + 0.693·21-s + 1.25i·23-s − 0.989·25-s − 0.398i·27-s − 0.367·29-s − 1.30i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(1.601762445\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.601762445\) |
| \(L(6)\) |
|
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 | \( 1 - 365. iT - 5.90e4T^{2} \) |
| 5 | \( 1 + 321.T + 9.76e6T^{2} \) |
| 7 | \( 1 + 7.74e3iT - 2.82e8T^{2} \) |
| 11 | \( 1 - 6.30e4iT - 2.59e10T^{2} \) |
| 13 | \( 1 - 5.88e5T + 1.37e11T^{2} \) |
| 17 | \( 1 + 1.10e6T + 2.01e12T^{2} \) |
| 19 | \( 1 + 1.40e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 - 8.10e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 + 7.53e6T + 4.20e14T^{2} \) |
| 31 | \( 1 + 3.74e7iT - 8.19e14T^{2} \) |
| 37 | \( 1 + 6.74e7T + 4.80e15T^{2} \) |
| 41 | \( 1 - 1.94e7T + 1.34e16T^{2} \) |
| 43 | \( 1 + 3.98e7iT - 2.16e16T^{2} \) |
| 47 | \( 1 + 3.13e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 - 2.99e8T + 1.74e17T^{2} \) |
| 59 | \( 1 + 9.75e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 - 5.95e8T + 7.13e17T^{2} \) |
| 67 | \( 1 + 1.47e9iT - 1.82e18T^{2} \) |
| 71 | \( 1 + 2.76e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 2.82e9T + 4.29e18T^{2} \) |
| 79 | \( 1 - 3.76e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 + 6.30e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 - 6.15e9T + 3.11e19T^{2} \) |
| 97 | \( 1 - 9.73e9T + 7.37e19T^{2} \) |
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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
−10.24646688720867482479011520918, −9.377886565698411437073911199001, −8.635590644744026370165726221503, −7.39186633193264297182489338669, −6.08909924570331147172800677786, −5.06191927120511995884384121014, −3.97236981137057577073173001603, −3.55800377775332593458302421287, −1.91026639808675586192785982353, −0.36421266839978025269471693272,
0.865990932141344378411733290936, 1.69696593163471817885638983067, 2.74002563199877550948929450686, 4.03323084052206393474288405097, 5.70561918881187078522485814792, 6.34587458245966603775000172254, 7.25642019686462692406780739161, 8.406208461331163851127614703164, 8.775257223128364930432300275707, 10.44969764309569516382694168972
