| L(s) = 1 | − 99.9·3-s − 610i·5-s − 1.39e3i·7-s + 3.42e3·9-s − 1.84e4·11-s − 5.47e3i·13-s + 6.09e4i·15-s + 7.30e4·17-s + 1.94e4·19-s + 1.39e5i·21-s + 2.37e5i·23-s + 1.85e4·25-s + 3.13e5·27-s − 1.28e5i·29-s − 6.79e4i·31-s + ⋯ |
| L(s) = 1 | − 1.23·3-s − 0.976i·5-s − 0.582i·7-s + 0.521·9-s − 1.26·11-s − 0.191i·13-s + 1.20i·15-s + 0.875·17-s + 0.149·19-s + 0.718i·21-s + 0.847i·23-s + 0.0474·25-s + 0.589·27-s − 0.181i·29-s − 0.0735i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.8773902364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8773902364\) |
| \(L(5)\) |
|
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 | \( 1 + 99.9T + 6.56e3T^{2} \) |
| 5 | \( 1 + 610iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 1.39e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 1.84e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + 5.47e3iT - 8.15e8T^{2} \) |
| 17 | \( 1 - 7.30e4T + 6.97e9T^{2} \) |
| 19 | \( 1 - 1.94e4T + 1.69e10T^{2} \) |
| 23 | \( 1 - 2.37e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 1.28e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 6.79e4iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 3.47e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 2.14e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + 5.92e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 7.62e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 8.24e5iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 3.72e6T + 1.46e14T^{2} \) |
| 61 | \( 1 + 1.47e7iT - 1.91e14T^{2} \) |
| 67 | \( 1 + 1.52e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 1.19e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 5.72e6T + 8.06e14T^{2} \) |
| 79 | \( 1 - 3.59e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 5.19e7T + 2.25e15T^{2} \) |
| 89 | \( 1 - 8.33e7T + 3.93e15T^{2} \) |
| 97 | \( 1 - 1.20e8T + 7.83e15T^{2} \) |
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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
−10.49990227843135655180263861792, −9.802423280759451623917092007922, −8.401765298321495865247861992403, −7.55048903423885961416800616489, −6.27462971809906186652444149985, −5.22961083214542580409781357652, −4.79914844610955696274167345929, −3.20494915335211756313711372669, −1.37007022894177337222404099398, −0.48517083897071666792839321961,
0.49670942186056966572309288554, 2.21768741122885016240363351138, 3.29421851909459324524517000417, 4.96861525798825246603678784058, 5.67295804813667664672790733388, 6.60246988791628375479050010079, 7.55199525783005403177197091128, 8.776853810765166589830615027498, 10.32667366396890183363638952350, 10.55365059302029412355707488305
