| L(s) = 1 | − 7.05i·3-s − 9.50·5-s − 6.49i·7-s + 31.2·9-s − 128. i·11-s − 220.·13-s + 67.0i·15-s − 382.·17-s − 77.3i·19-s − 45.8·21-s + 951. i·23-s − 534.·25-s − 791. i·27-s − 561.·29-s + 321. i·31-s + ⋯ |
| L(s) = 1 | − 0.784i·3-s − 0.380·5-s − 0.132i·7-s + 0.385·9-s − 1.06i·11-s − 1.30·13-s + 0.298i·15-s − 1.32·17-s − 0.214i·19-s − 0.103·21-s + 1.79i·23-s − 0.855·25-s − 1.08i·27-s − 0.667·29-s + 0.334i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(-0.624077i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(-0.624077i\) |
| \(L(3)\) |
|
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 | \( 1 + 7.05iT - 81T^{2} \) |
| 5 | \( 1 + 9.50T + 625T^{2} \) |
| 7 | \( 1 + 6.49iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 128. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 220.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 382.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 77.3iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 951. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 561.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 321. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.09e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 437.T + 2.82e6T^{2} \) |
| 43 | \( 1 + 2.55e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 3.83e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.52e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 1.75e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 4.76e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 6.91e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 337. iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 4.23e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 1.10e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 3.97e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 5.52e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 4.73e3T + 8.85e7T^{2} \) |
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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
−12.09190908741345345807142163184, −11.34469973673742239339468872611, −10.04656128112015865894481667791, −8.801944318092928672874707938640, −7.57902741908471434987478853534, −6.86176991674984336803197470474, −5.37219671491835483622344158329, −3.78884031721314120859374095778, −2.02438160819656456912069754430, −0.24540354999262541751768330095,
2.28686998478896560241873101043, 4.17050158978599183144591757648, 4.86925375971031032269412550189, 6.68364385721652978637180688530, 7.75802791546166558179421590579, 9.200260818858760116281972244666, 9.978290757800185215172540884706, 10.94787277507185262192635983339, 12.19833689464187438270379921126, 12.92448552504918091839484033074
