L(s) = 1 | + 2.15i·2-s + (2.77 − 1.60i)3-s − 0.635·4-s + (−0.468 + 0.270i)5-s + (3.45 + 5.98i)6-s + (−7.68 − 4.43i)7-s + 7.24i·8-s + (0.644 − 1.11i)9-s + (−0.582 − 1.00i)10-s − 6.16·11-s + (−1.76 + 1.01i)12-s + (6.81 − 11.8i)13-s + (9.55 − 16.5i)14-s + (−0.868 + 1.50i)15-s − 18.1·16-s + (8.71 − 15.1i)17-s + ⋯ |
L(s) = 1 | + 1.07i·2-s + (0.925 − 0.534i)3-s − 0.158·4-s + (−0.0937 + 0.0541i)5-s + (0.575 + 0.996i)6-s + (−1.09 − 0.633i)7-s + 0.905i·8-s + (0.0715 − 0.123i)9-s + (−0.0582 − 0.100i)10-s − 0.560·11-s + (−0.147 + 0.0849i)12-s + (0.524 − 0.908i)13-s + (0.682 − 1.18i)14-s + (−0.0578 + 0.100i)15-s − 1.13·16-s + (0.512 − 0.888i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.20689 + 0.564524i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20689 + 0.564524i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 + (-30.5 - 30.3i)T \) |
good | 2 | \( 1 - 2.15iT - 4T^{2} \) |
| 3 | \( 1 + (-2.77 + 1.60i)T + (4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (0.468 - 0.270i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (7.68 + 4.43i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + 6.16T + 121T^{2} \) |
| 13 | \( 1 + (-6.81 + 11.8i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (-8.71 + 15.1i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (1.57 - 0.912i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-12.8 - 22.2i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-17.4 - 10.0i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (27.1 + 47.0i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-8.14 + 4.70i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 48.7T + 1.68e3T^{2} \) |
| 47 | \( 1 + 56.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + (-27.2 - 47.1i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 - 28.4T + 3.48e3T^{2} \) |
| 61 | \( 1 + (28.7 + 16.5i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-1.46 - 2.54i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (102. + 58.9i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-112. - 64.9i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-59.5 + 103. i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (46.8 + 81.2i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (127. - 73.7i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 66.7T + 9.40e3T^{2} \) |
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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
−15.85906437412010340344068115931, −14.85212891461712045366029878085, −13.66766312648611497057894994796, −13.00376228916113041503504941821, −11.05146526584975901921004400154, −9.400459613703445597411576082639, −7.88192419390825148559171641644, −7.26275808855017233869105248852, −5.70591089279072168371953428010, −3.05048057921254500990940456716,
2.63807099224144248612488579381, 3.85142171110210299077940350168, 6.44587416223577296971031506720, 8.596099335342314917933289410680, 9.602017276054941061227449750597, 10.61037052729238759763150754036, 12.10867232707208552749961845530, 12.96713227657228327294781032593, 14.36480506018532032708356003041, 15.63971598429679273357463521911