| L(s) = 1 | + 17.0·2-s + (−45.4 − 78.6i)3-s + 164.·4-s + (−83.8 − 145. i)5-s + (−776. − 1.34e3i)6-s + (−130. + 226. i)7-s + 619.·8-s + (−3.03e3 + 5.24e3i)9-s + (−1.43e3 − 2.48e3i)10-s + 1.96e3·11-s + (−7.45e3 − 1.29e4i)12-s + (2.21e3 − 3.83e3i)13-s + (−2.23e3 + 3.86e3i)14-s + (−7.61e3 + 1.31e4i)15-s − 1.04e4·16-s + (−2.36e3 + 4.09e3i)17-s + ⋯ |
| L(s) = 1 | + 1.51·2-s + (−0.971 − 1.68i)3-s + 1.28·4-s + (−0.300 − 0.519i)5-s + (−1.46 − 2.54i)6-s + (−0.143 + 0.249i)7-s + 0.427·8-s + (−1.38 + 2.40i)9-s + (−0.453 − 0.785i)10-s + 0.444·11-s + (−1.24 − 2.15i)12-s + (0.279 − 0.484i)13-s + (−0.217 + 0.376i)14-s + (−0.582 + 1.00i)15-s − 0.636·16-s + (−0.116 + 0.202i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0402i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0367931 - 1.82741i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0367931 - 1.82741i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (3.98e5 + 3.36e5i)T \) |
| good | 2 | \( 1 - 17.0T + 128T^{2} \) |
| 3 | \( 1 + (45.4 + 78.6i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (83.8 + 145. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (130. - 226. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 - 1.96e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (-2.21e3 + 3.83e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (2.36e3 - 4.09e3i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (2.32e4 + 4.02e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (2.20e4 + 3.82e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-2.41e4 + 4.18e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (1.08e5 + 1.88e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-2.67e5 - 4.63e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 - 5.48e5T + 1.94e11T^{2} \) |
| 47 | \( 1 + 4.33e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (8.71e5 + 1.51e6i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 - 2.74e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + (-4.77e4 + 8.26e4i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.04e6 - 1.80e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (1.35e6 - 2.35e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-6.89e5 + 1.19e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-3.14e6 + 5.45e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-1.57e6 - 2.72e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (3.29e6 + 5.70e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 - 3.05e6T + 8.07e13T^{2} \) |
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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
−13.27840548205169222337567866851, −12.90427651397319690724708728015, −11.94373968335436645756562624282, −11.16439556404137313872425602183, −8.345007275430897633565201069277, −6.76687496208568464319596260801, −5.94993064209947230152673212536, −4.66226194594873696805552196142, −2.38439150122726478641599595551, −0.51163447191256174621317109997,
3.49402856248670023731540747652, 4.19297279359756909020526788789, 5.50847910268680710313421203958, 6.55274283213041399322976463846, 9.291847052109049976667796254389, 10.69518280605450449283264902600, 11.45430233154032164384844627318, 12.49313000404506362025065670835, 14.31160710243927007142965309329, 14.85238253982376663787272397293
