| L(s) = 1 | − 63.9·3-s + 217. i·5-s + 1.72e3i·7-s − 2.47e3·9-s + 5.95e3·11-s + 4.91e4i·13-s − 1.39e4i·15-s − 5.56e4·17-s − 2.02e5·19-s − 1.10e5i·21-s + 2.09e5i·23-s + 3.43e5·25-s + 5.77e5·27-s − 3.08e3i·29-s + 1.36e6i·31-s + ⋯ |
| L(s) = 1 | − 0.789·3-s + 0.348i·5-s + 0.718i·7-s − 0.377·9-s + 0.406·11-s + 1.71i·13-s − 0.274i·15-s − 0.666·17-s − 1.55·19-s − 0.567i·21-s + 0.748i·23-s + 0.878·25-s + 1.08·27-s − 0.00435i·29-s + 1.47i·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
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.5566902647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5566902647\) |
| \(L(5)\) |
|
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 | \( 1 + 63.9T + 6.56e3T^{2} \) |
| 5 | \( 1 - 217. iT - 3.90e5T^{2} \) |
| 7 | \( 1 - 1.72e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 5.95e3T + 2.14e8T^{2} \) |
| 13 | \( 1 - 4.91e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 5.56e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 2.02e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 2.09e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 3.08e3iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 1.36e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 1.31e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 4.54e6T + 7.98e12T^{2} \) |
| 43 | \( 1 - 3.48e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 7.23e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 2.48e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 1.48e7T + 1.46e14T^{2} \) |
| 61 | \( 1 + 4.13e6iT - 1.91e14T^{2} \) |
| 67 | \( 1 + 6.04e6T + 4.06e14T^{2} \) |
| 71 | \( 1 - 4.20e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 9.00e6T + 8.06e14T^{2} \) |
| 79 | \( 1 + 6.81e6iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 5.81e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + 7.82e7T + 3.93e15T^{2} \) |
| 97 | \( 1 - 6.01e7T + 7.83e15T^{2} \) |
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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
−11.31673212874327933825551810124, −10.46392719798669770062668304631, −9.101309642445791669338657688937, −8.570313830181558008309882279314, −6.79721655103902998772934999528, −6.42346170354979851749221679762, −5.20281212255798892464587673863, −4.17351601621184089606396847303, −2.67234474753445407118682968698, −1.52213963821763139380834915807,
0.18618112366940577662160346236, 0.75168683266135777132352893554, 2.43258438073114461439701432174, 3.88696710670083176763472675318, 4.94911433736512111365798797020, 5.94551863854427842962196731562, 6.83727054216234691349954386146, 8.111157015473754213348645357678, 8.913724373836327574597831800712, 10.45858458729156754521141640269
