| L(s) = 1 | − 6.28·2-s + (10.0 + 17.4i)3-s + 7.51·4-s + (38.9 + 67.5i)5-s + (−63.3 − 109. i)6-s + (4.83 − 8.37i)7-s + 153.·8-s + (−81.3 + 140. i)9-s + (−245. − 424. i)10-s − 186.·11-s + (75.7 + 131. i)12-s + (−333. + 577. i)13-s + (−30.3 + 52.6i)14-s + (−785. + 1.36e3i)15-s − 1.20e3·16-s + (−506. + 877. i)17-s + ⋯ |
| L(s) = 1 | − 1.11·2-s + (0.646 + 1.11i)3-s + 0.234·4-s + (0.697 + 1.20i)5-s + (−0.718 − 1.24i)6-s + (0.0372 − 0.0645i)7-s + 0.850·8-s + (−0.334 + 0.580i)9-s + (−0.774 − 1.34i)10-s − 0.465·11-s + (0.151 + 0.262i)12-s + (−0.547 + 0.947i)13-s + (−0.0414 + 0.0717i)14-s + (−0.901 + 1.56i)15-s − 1.17·16-s + (−0.425 + 0.736i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.838 - 0.545i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.838 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.278451 + 0.937997i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.278451 + 0.937997i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-3.01e3 - 1.17e4i)T \) |
| good | 2 | \( 1 + 6.28T + 32T^{2} \) |
| 3 | \( 1 + (-10.0 - 17.4i)T + (-121.5 + 210. i)T^{2} \) |
| 5 | \( 1 + (-38.9 - 67.5i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-4.83 + 8.37i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + 186.T + 1.61e5T^{2} \) |
| 13 | \( 1 + (333. - 577. i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + (506. - 877. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (510. + 884. i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (359. + 622. i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-2.47e3 + 4.28e3i)T + (-1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + (1.88e3 + 3.26e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-7.59e3 - 1.31e4i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + 1.16e4T + 1.15e8T^{2} \) |
| 47 | \( 1 - 1.04e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + (1.14e4 + 1.98e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 - 3.72e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + (5.92e3 - 1.02e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-2.97e4 - 5.15e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-4.85e3 + 8.40e3i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (-3.85e4 + 6.67e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (4.50e4 - 7.80e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-8.00e3 - 1.38e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-3.79e4 - 6.57e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + 8.09e4T + 8.58e9T^{2} \) |
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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
−15.39807083794007853730844607192, −14.49731686610205667212805948015, −13.48571680837252761901230589980, −11.10517931943045402249731633081, −10.12887899586026789693249843908, −9.564471772542704868534064367399, −8.307404779016906227977160350228, −6.70827021804433836726103958305, −4.38288161436444154866526030522, −2.43932726295227813972819567919,
0.71956114971248323703577410171, 2.07054845272405804724290691882, 5.18314970343256085115200788671, 7.29375401041344177639779135644, 8.321951459896005875554453711257, 9.144369245422849399063655200499, 10.39697219554440223198364209893, 12.47429438872053852799907018627, 13.18199797067718281324787729700, 14.13191299320619247585716300443
