| L(s) = 1 | + (−13.7 − 17.3i)2-s + (−134. − 20.3i)3-s + (4.96 − 21.7i)4-s + (1.09e3 + 746. i)5-s + (1.50e3 + 2.61e3i)6-s + (2.83e3 − 4.91e3i)7-s + (−1.06e4 + 5.13e3i)8-s + (−1.01e3 − 314. i)9-s + (−2.19e3 − 2.92e4i)10-s + (−9.96e3 − 4.36e4i)11-s + (−1.11e3 + 2.83e3i)12-s + (1.33e4 − 1.78e5i)13-s + (−1.24e5 + 1.87e4i)14-s + (−1.32e5 − 1.22e5i)15-s + (2.25e5 + 1.08e5i)16-s + (−4.64e5 + 3.16e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.609 − 0.764i)2-s + (−0.961 − 0.144i)3-s + (0.00969 − 0.0424i)4-s + (0.783 + 0.533i)5-s + (0.475 + 0.823i)6-s + (0.446 − 0.773i)7-s + (−0.919 + 0.442i)8-s + (−0.0517 − 0.0159i)9-s + (−0.0692 − 0.924i)10-s + (−0.205 − 0.899i)11-s + (−0.0154 + 0.0394i)12-s + (0.130 − 1.73i)13-s + (−0.863 + 0.130i)14-s + (−0.675 − 0.627i)15-s + (0.859 + 0.414i)16-s + (−1.34 + 0.919i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.146331 + 0.174366i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.146331 + 0.174366i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (2.17e7 - 5.40e6i)T \) |
| good | 2 | \( 1 + (13.7 + 17.3i)T + (-113. + 499. i)T^{2} \) |
| 3 | \( 1 + (134. + 20.3i)T + (1.88e4 + 5.80e3i)T^{2} \) |
| 5 | \( 1 + (-1.09e3 - 746. i)T + (7.13e5 + 1.81e6i)T^{2} \) |
| 7 | \( 1 + (-2.83e3 + 4.91e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (9.96e3 + 4.36e4i)T + (-2.12e9 + 1.02e9i)T^{2} \) |
| 13 | \( 1 + (-1.33e4 + 1.78e5i)T + (-1.04e10 - 1.58e9i)T^{2} \) |
| 17 | \( 1 + (4.64e5 - 3.16e5i)T + (4.33e10 - 1.10e11i)T^{2} \) |
| 19 | \( 1 + (-5.50e5 + 1.69e5i)T + (2.66e11 - 1.81e11i)T^{2} \) |
| 23 | \( 1 + (3.25e5 - 3.01e5i)T + (1.34e11 - 1.79e12i)T^{2} \) |
| 29 | \( 1 + (2.60e6 - 3.93e5i)T + (1.38e13 - 4.27e12i)T^{2} \) |
| 31 | \( 1 + (1.51e5 - 3.86e5i)T + (-1.93e13 - 1.79e13i)T^{2} \) |
| 37 | \( 1 + (-5.74e6 - 9.95e6i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 + (2.46e6 + 3.08e6i)T + (-7.28e13 + 3.19e14i)T^{2} \) |
| 47 | \( 1 + (1.22e7 - 5.37e7i)T + (-1.00e15 - 4.85e14i)T^{2} \) |
| 53 | \( 1 + (5.60e6 + 7.48e7i)T + (-3.26e15 + 4.91e14i)T^{2} \) |
| 59 | \( 1 + (1.36e8 + 6.55e7i)T + (5.40e15 + 6.77e15i)T^{2} \) |
| 61 | \( 1 + (-1.18e7 - 3.01e7i)T + (-8.57e15 + 7.95e15i)T^{2} \) |
| 67 | \( 1 + (2.32e6 - 7.17e5i)T + (2.24e16 - 1.53e16i)T^{2} \) |
| 71 | \( 1 + (-1.42e8 - 1.31e8i)T + (3.42e15 + 4.57e16i)T^{2} \) |
| 73 | \( 1 + (1.72e7 - 2.30e8i)T + (-5.82e16 - 8.77e15i)T^{2} \) |
| 79 | \( 1 + (2.77e8 - 4.80e8i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-2.03e8 - 3.06e7i)T + (1.78e17 + 5.51e16i)T^{2} \) |
| 89 | \( 1 + (1.13e9 + 1.70e8i)T + (3.34e17 + 1.03e17i)T^{2} \) |
| 97 | \( 1 + (7.49e7 + 3.28e8i)T + (-6.84e17 + 3.29e17i)T^{2} \) |
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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
−12.94407580829913694266128688149, −11.27411263997603520779196090727, −10.87452378100892215610622123468, −9.936504563337421544301712179336, −8.274121314374697875733785519644, −6.34102984657940256060992900895, −5.44896485043575512154874520050, −2.95320450840331997396082944899, −1.24162985121812222589771864870, −0.11607077896719439708379508461,
2.02636659308674472393823907669, 4.79216244855380053507270518957, 5.94582277973662959424481303589, 7.12650040876246766512779244637, 8.827523980813710179408400397556, 9.543801585967246917003004916624, 11.44530281693204898596596419056, 12.17278754128642884110181286900, 13.70638038265855613775562575733, 15.21106976233139802675282040374
