L(s) = 1 | + 10.2·2-s + (56.7 + 57.8i)3-s − 151.·4-s + (580. + 591. i)6-s + 3.44e3i·7-s − 4.16e3·8-s + (−122. + 6.55e3i)9-s − 7.12e3i·11-s + (−8.58e3 − 8.74e3i)12-s − 4.13e4i·13-s + 3.52e4i·14-s − 3.95e3·16-s − 1.19e5·17-s + (−1.25e3 + 6.71e4i)18-s + 8.62e4·19-s + ⋯ |
L(s) = 1 | + 0.639·2-s + (0.700 + 0.713i)3-s − 0.590·4-s + (0.448 + 0.456i)6-s + 1.43i·7-s − 1.01·8-s + (−0.0187 + 0.999i)9-s − 0.486i·11-s + (−0.413 − 0.421i)12-s − 1.44i·13-s + 0.918i·14-s − 0.0602·16-s − 1.43·17-s + (−0.0119 + 0.639i)18-s + 0.662·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.129463 - 0.822131i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.129463 - 0.822131i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-56.7 - 57.8i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 10.2T + 256T^{2} \) |
| 7 | \( 1 - 3.44e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 7.12e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 4.13e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 1.19e5T + 6.97e9T^{2} \) |
| 19 | \( 1 - 8.62e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + 3.17e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + 5.98e4iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 1.02e6T + 8.52e11T^{2} \) |
| 37 | \( 1 - 8.77e5iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 1.55e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 2.56e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 8.98e6T + 2.38e13T^{2} \) |
| 53 | \( 1 - 6.22e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 3.96e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 5.63e5T + 1.91e14T^{2} \) |
| 67 | \( 1 - 1.34e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 3.56e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 9.70e5iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 2.78e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 2.25e7T + 2.25e15T^{2} \) |
| 89 | \( 1 - 5.73e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 3.31e7iT - 7.83e15T^{2} \) |
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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
−13.50292929557238173072806522193, −12.69434365329559006855174312980, −11.38977283219665711193790464269, −9.911001906226168448918467872521, −8.901894057332139961709794191423, −8.180531173834879204949029757419, −5.85414930167452293129140856072, −5.00110372214406643702417412448, −3.56157906085279772344306747434, −2.49650619205964231794364065176,
0.18244278057227168385802431761, 1.83512210176518526318865434264, 3.67096884895898160185760400354, 4.49778152775838380534969461987, 6.47376039200318584171746810522, 7.43163159280478973678034655892, 8.832127750988716647611199758918, 9.795685205045791667592082077450, 11.47082271794563406977715578849, 12.65552170761201155682962257296