| L(s) = 1 | + (5.65 − 9.79i)2-s + (−21.7 + 12.5i)3-s + (−63.9 − 110. i)4-s + (−492. − 284. i)5-s + 283. i·6-s + (−1.74e3 − 1.65e3i)7-s − 1.44e3·8-s + (−2.96e3 + 5.13e3i)9-s + (−5.56e3 + 3.21e3i)10-s + (−2.95e3 − 5.11e3i)11-s + (2.78e3 + 1.60e3i)12-s − 2.51e4i·13-s + (−2.60e4 + 7.70e3i)14-s + 1.42e4·15-s + (−8.19e3 + 1.41e4i)16-s + (1.04e5 − 6.02e4i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.268 + 0.154i)3-s + (−0.249 − 0.433i)4-s + (−0.787 − 0.454i)5-s + 0.219i·6-s + (−0.725 − 0.688i)7-s − 0.353·8-s + (−0.452 + 0.782i)9-s + (−0.556 + 0.321i)10-s + (−0.201 − 0.349i)11-s + (0.134 + 0.0774i)12-s − 0.881i·13-s + (−0.678 + 0.200i)14-s + 0.281·15-s + (−0.125 + 0.216i)16-s + (1.24 − 0.720i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.160i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0611367 - 0.758623i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0611367 - 0.758623i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-5.65 + 9.79i)T \) |
| 7 | \( 1 + (1.74e3 + 1.65e3i)T \) |
| good | 3 | \( 1 + (21.7 - 12.5i)T + (3.28e3 - 5.68e3i)T^{2} \) |
| 5 | \( 1 + (492. + 284. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (2.95e3 + 5.11e3i)T + (-1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + 2.51e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (-1.04e5 + 6.02e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-8.33e4 - 4.81e4i)T + (8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-1.69e4 + 2.93e4i)T + (-3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + 1.22e6T + 5.00e11T^{2} \) |
| 31 | \( 1 + (-4.55e4 + 2.62e4i)T + (4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (9.87e5 - 1.71e6i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + 1.61e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 3.53e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (4.62e6 + 2.66e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (4.32e6 + 7.48e6i)T + (-3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-1.61e7 + 9.34e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (2.03e7 + 1.17e7i)T + (9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-1.83e6 - 3.18e6i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 - 4.94e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-2.86e6 + 1.65e6i)T + (4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (3.73e7 - 6.46e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + 2.45e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (3.07e7 + 1.77e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 - 7.41e7iT - 7.83e15T^{2} \) |
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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
−16.83890819061160585782396272160, −15.85592854181398999627214430412, −14.00772749106946598217077350586, −12.70354950683455442281772458123, −11.33936849119047215500249775452, −9.986324792065270230801403688245, −7.86032798987605617753489142822, −5.33662490308688030010680763013, −3.43444791805540276375881814571, −0.41162999514634069194585533385,
3.49135773693973568131115411674, 5.87364600215280282889458498943, 7.36795829568930100709298485568, 9.272249052469462370099132832183, 11.58319607216207350865383734066, 12.65165734628022635174841873068, 14.53869636416646685551488106925, 15.52201593952874659907572495609, 16.78694807671579971258935334113, 18.30515306690971485055632637518
