L(s) = 1 | − 3.03·2-s + (−79.6 − 14.6i)3-s − 246.·4-s + (241. + 44.3i)6-s + 59.7i·7-s + 1.52e3·8-s + (6.13e3 + 2.32e3i)9-s − 2.18e4i·11-s + (1.96e4 + 3.60e3i)12-s − 3.84e4i·13-s − 181. i·14-s + 5.85e4·16-s − 3.61e4·17-s + (−1.86e4 − 7.06e3i)18-s + 1.36e5·19-s + ⋯ |
L(s) = 1 | − 0.189·2-s + (−0.983 − 0.180i)3-s − 0.964·4-s + (0.186 + 0.0341i)6-s + 0.0248i·7-s + 0.372·8-s + (0.934 + 0.354i)9-s − 1.49i·11-s + (0.948 + 0.173i)12-s − 1.34i·13-s − 0.00471i·14-s + 0.893·16-s − 0.432·17-s + (−0.177 − 0.0672i)18-s + 1.04·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.799 - 0.601i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.799 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0357454 + 0.106982i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0357454 + 0.106982i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (79.6 + 14.6i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 3.03T + 256T^{2} \) |
| 7 | \( 1 - 59.7iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 2.18e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 3.84e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 3.61e4T + 6.97e9T^{2} \) |
| 19 | \( 1 - 1.36e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + 2.35e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 8.96e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 6.80e4T + 8.52e11T^{2} \) |
| 37 | \( 1 - 6.65e4iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 3.76e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 2.03e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 5.93e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + 1.31e7T + 6.22e13T^{2} \) |
| 59 | \( 1 - 1.58e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 3.51e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + 1.39e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 1.51e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 3.93e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 3.15e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 4.88e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + 1.09e8iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.16e8iT - 7.83e15T^{2} \) |
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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
−12.26695197293387433974497223256, −10.99317060582233421133413526028, −10.13169167137704741161343867045, −8.751677771095687657492634594637, −7.62180572406165272870849898806, −5.92458399241920909926605459363, −5.10416521456531127375099649931, −3.49131916920707632686606603551, −1.01870488916351611012484876158, −0.05733503602289319729313923547,
1.57812323861695616456451819196, 4.16805003531442442697288456650, 4.89365903828101080349923827250, 6.43362573759309250564215478039, 7.72032646951668573151216519054, 9.467501481909021994410765649527, 9.901704394805658096018865923663, 11.42764180208308206876573115045, 12.33319875985444554917315900491, 13.41423688219159996737546103183