| L(s) = 1 | + 2.64i·2-s + (−0.554 − 0.320i)3-s + 57.0·4-s + (14.6 + 8.48i)5-s + (0.846 − 1.46i)6-s + (131. − 76.0i)7-s + 320. i·8-s + (−364. − 630. i)9-s + (−22.4 + 38.8i)10-s + 1.69e3·11-s + (−31.6 − 18.2i)12-s + (892. + 1.54e3i)13-s + (201. + 348. i)14-s + (−5.43 − 9.40i)15-s + 2.80e3·16-s + (1.21e3 + 2.09e3i)17-s + ⋯ |
| L(s) = 1 | + 0.330i·2-s + (−0.0205 − 0.0118i)3-s + 0.890·4-s + (0.117 + 0.0678i)5-s + (0.00392 − 0.00678i)6-s + (0.384 − 0.221i)7-s + 0.625i·8-s + (−0.499 − 0.865i)9-s + (−0.0224 + 0.0388i)10-s + 1.27·11-s + (−0.0182 − 0.0105i)12-s + (0.406 + 0.703i)13-s + (0.0733 + 0.127i)14-s + (−0.00160 − 0.00278i)15-s + 0.684·16-s + (0.246 + 0.426i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.402i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.915 - 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.25585 + 0.474136i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.25585 + 0.474136i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (3.81e4 + 6.97e4i)T \) |
| good | 2 | \( 1 - 2.64iT - 64T^{2} \) |
| 3 | \( 1 + (0.554 + 0.320i)T + (364.5 + 631. i)T^{2} \) |
| 5 | \( 1 + (-14.6 - 8.48i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 7 | \( 1 + (-131. + 76.0i)T + (5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 - 1.69e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + (-892. - 1.54e3i)T + (-2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 + (-1.21e3 - 2.09e3i)T + (-1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-8.14e3 - 4.70e3i)T + (2.35e7 + 4.07e7i)T^{2} \) |
| 23 | \( 1 + (2.99e3 - 5.19e3i)T + (-7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-3.63e3 + 2.09e3i)T + (2.97e8 - 5.15e8i)T^{2} \) |
| 31 | \( 1 + (-1.17e4 + 2.03e4i)T + (-4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + (4.41e4 + 2.54e4i)T + (1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + 6.83e4T + 4.75e9T^{2} \) |
| 47 | \( 1 + 1.43e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + (-4.02e3 + 6.97e3i)T + (-1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 - 2.70e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + (-1.95e5 + 1.12e5i)T + (2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (6.97e4 - 1.20e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + (5.01e5 - 2.89e5i)T + (6.40e10 - 1.10e11i)T^{2} \) |
| 73 | \( 1 + (-3.57e5 + 2.06e5i)T + (7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (3.87e5 + 6.71e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (2.34e5 - 4.06e5i)T + (-1.63e11 - 2.83e11i)T^{2} \) |
| 89 | \( 1 + (5.11e5 + 2.95e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 + 1.06e6T + 8.32e11T^{2} \) |
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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
−14.69784988665128775779879501109, −14.02564614309540257587924934016, −11.97467663937300106918125743358, −11.51734646862143666778225962568, −9.868908810649277403167269649729, −8.387282676244572120792319357239, −6.88785994062937501569428136572, −5.88460677969532109154710596744, −3.64140857416301397120710975872, −1.52694279763287481352335941273,
1.45075063364234644928626121968, 3.13981257756272704237461067606, 5.33268285467617490617243987389, 6.87070034916201730553972769786, 8.314842349064521335129909880910, 9.914978389099296433183547103764, 11.26257719027802954990085649842, 11.87907145363092629225022457686, 13.43641192907225825191683418404, 14.62514650670263630985114333812
