L(s) = 1 | − 1.70i·3-s + (2.17 − 0.539i)5-s − 2.63i·7-s + 0.0783·9-s + 5.41·11-s + 6.34i·13-s + (−0.921 − 3.70i)15-s − 3.41i·17-s − 3.26·19-s − 4.49·21-s + 1.36i·23-s + (4.41 − 2.34i)25-s − 5.26i·27-s − 2·29-s − 4.68·31-s + ⋯ |
L(s) = 1 | − 0.986i·3-s + (0.970 − 0.241i)5-s − 0.994i·7-s + 0.0261·9-s + 1.63·11-s + 1.75i·13-s + (−0.237 − 0.957i)15-s − 0.829i·17-s − 0.748·19-s − 0.981·21-s + 0.285i·23-s + (0.883 − 0.468i)25-s − 1.01i·27-s − 0.371·29-s − 0.840·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.50686 - 1.17828i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.50686 - 1.17828i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.17 + 0.539i)T \) |
good | 3 | \( 1 + 1.70iT - 3T^{2} \) |
| 7 | \( 1 + 2.63iT - 7T^{2} \) |
| 11 | \( 1 - 5.41T + 11T^{2} \) |
| 13 | \( 1 - 6.34iT - 13T^{2} \) |
| 17 | \( 1 + 3.41iT - 17T^{2} \) |
| 19 | \( 1 + 3.26T + 19T^{2} \) |
| 23 | \( 1 - 1.36iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 4.68T + 31T^{2} \) |
| 37 | \( 1 - 5.75iT - 37T^{2} \) |
| 41 | \( 1 + 7.75T + 41T^{2} \) |
| 43 | \( 1 - 4.44iT - 43T^{2} \) |
| 47 | \( 1 + 4.78iT - 47T^{2} \) |
| 53 | \( 1 - 1.65iT - 53T^{2} \) |
| 59 | \( 1 - 3.26T + 59T^{2} \) |
| 61 | \( 1 + 2.49T + 61T^{2} \) |
| 67 | \( 1 + 7.86iT - 67T^{2} \) |
| 71 | \( 1 + 6.15T + 71T^{2} \) |
| 73 | \( 1 + 13.5iT - 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 14.9iT - 83T^{2} \) |
| 89 | \( 1 - 8.52T + 89T^{2} \) |
| 97 | \( 1 - 4.58iT - 97T^{2} \) |
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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
−10.28418202924388796173496659025, −9.400878632271644383817764608259, −8.844551501760473184137579360482, −7.45515996808005722640010121383, −6.67198336643281702045211255376, −6.38843382021056645443949287934, −4.76528246467753403098928198957, −3.85168577274616627528014900771, −1.97456931574189927582470939468, −1.25678848664019354551084836957,
1.76714006595715220593271578996, 3.15912371916785394519871354883, 4.17698500981928025281958118525, 5.46364143948528066999107001517, 5.96138699379922525405491832147, 7.06644250512636940827475457903, 8.591187303622340888009112286047, 9.053013229049960321478545216173, 9.991395470807294461559911605241, 10.48809798627960219600247196959