L(s) = 1 | + 20i·3-s + 24i·7-s − 157·9-s + 124·11-s + 478i·13-s + 1.19e3i·17-s − 3.04e3·19-s − 480·21-s + 184i·23-s + 1.72e3i·27-s + 3.28e3·29-s − 5.72e3·31-s + 2.48e3i·33-s − 1.03e4i·37-s − 9.56e3·39-s + ⋯ |
L(s) = 1 | + 1.28i·3-s + 0.185i·7-s − 0.646·9-s + 0.308·11-s + 0.784i·13-s + 1.00i·17-s − 1.93·19-s − 0.237·21-s + 0.0725i·23-s + 0.454i·27-s + 0.724·29-s − 1.07·31-s + 0.396i·33-s − 1.24i·37-s − 1.00·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.8846103073\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8846103073\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 20iT - 243T^{2} \) |
| 7 | \( 1 - 24iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 124T + 1.61e5T^{2} \) |
| 13 | \( 1 - 478iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.19e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 3.04e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 184iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 3.28e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.72e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.03e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 8.88e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.18e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.36e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.16e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 1.68e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.84e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.55e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 3.19e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.88e3iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 4.45e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.73e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 7.19e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.88e4iT - 8.58e9T^{2} \) |
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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
−12.03572340098120601598174672203, −10.83012660504994142919042566429, −10.33429437644249312575827315683, −9.140979277202607640645187476456, −8.550444130005606340715493837899, −6.91489046709877375782035709337, −5.73842538736061835062800473690, −4.42902416785970411890503377463, −3.74843277682716594817605308180, −1.98498027830874329304439730333,
0.26108761409163315725902299614, 1.53897883558213299010803342532, 2.84593751831303045828783058062, 4.54273724968943560426844887269, 6.06211529109739476822646204276, 6.89938247224806911676175738282, 7.83869722148292122706082441747, 8.761513141881721603336813401091, 10.11132540666063241857693165950, 11.16993604996919892882464326307