| L(s) = 1 | + (−2.44 + 1.41i)2-s + (0.191 − 8.99i)3-s + (3.99 − 6.92i)4-s + (−11.9 + 6.92i)5-s + (12.2 + 22.3i)6-s + (−39.6 + 28.8i)7-s + 22.6i·8-s + (−80.9 − 3.43i)9-s + (19.5 − 33.9i)10-s + (−90.4 − 52.2i)11-s + (−61.5 − 37.3i)12-s − 131.·13-s + (56.2 − 126. i)14-s + (59.9 + 109. i)15-s + (−32.0 − 55.4i)16-s + (−140. − 81.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.0212 − 0.999i)3-s + (0.249 − 0.433i)4-s + (−0.479 + 0.276i)5-s + (0.340 + 0.619i)6-s + (−0.808 + 0.588i)7-s + 0.353i·8-s + (−0.999 − 0.0424i)9-s + (0.195 − 0.339i)10-s + (−0.747 − 0.431i)11-s + (−0.427 − 0.259i)12-s − 0.779·13-s + (0.287 − 0.646i)14-s + (0.266 + 0.485i)15-s + (−0.125 − 0.216i)16-s + (−0.485 − 0.280i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0208i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.00132239 - 0.126690i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00132239 - 0.126690i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.44 - 1.41i)T \) |
| 3 | \( 1 + (-0.191 + 8.99i)T \) |
| 7 | \( 1 + (39.6 - 28.8i)T \) |
| good | 5 | \( 1 + (11.9 - 6.92i)T + (312.5 - 541. i)T^{2} \) |
| 11 | \( 1 + (90.4 + 52.2i)T + (7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + 131.T + 2.85e4T^{2} \) |
| 17 | \( 1 + (140. + 81.0i)T + (4.17e4 + 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-123. - 213. i)T + (-6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (-518. + 299. i)T + (1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + 1.39e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + (-241. + 418. i)T + (-4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + (1.10e3 + 1.92e3i)T + (-9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 - 706. iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 976.T + 3.41e6T^{2} \) |
| 47 | \( 1 + (3.79e3 - 2.18e3i)T + (2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + (-3.70e3 - 2.13e3i)T + (3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (515. + 297. i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (2.87e3 + 4.97e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (3.19e3 - 5.53e3i)T + (-1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + 965. iT - 2.54e7T^{2} \) |
| 73 | \( 1 + (1.60e3 - 2.77e3i)T + (-1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-1.22e3 - 2.11e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 - 5.44e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + (9.39e3 - 5.42e3i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 97 | \( 1 + 9.79e3T + 8.85e7T^{2} \) |
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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
−14.86675224990471823067422085014, −13.45675194850707326509617976235, −12.30884900754137396135620780157, −11.13143414032763276566177206587, −9.497476218718122862298773622907, −8.128676087954377768081607399888, −7.06600227503673152601068265091, −5.75255878253378345791788615813, −2.66747433819975516826445714908, −0.093720448243654555504379379049,
3.17282743999163974700667674958, 4.81939432826681385381625821607, 7.12660127789193673601332739428, 8.682354688110484296798555969434, 9.866072596288911026634643575172, 10.71043554892854520461382226101, 12.06508769541931572745952665617, 13.38004795232350853617394672606, 15.08321485916421345267339022518, 15.98281251839015238914346682244
