| L(s) = 1 | + 1.72i·2-s + (2.56 + 1.55i)3-s + 1.03·4-s + (−1.01 − 1.75i)5-s + (−2.68 + 4.41i)6-s + (−1.46 − 2.53i)7-s + 8.66i·8-s + (4.13 + 7.99i)9-s + (3.02 − 1.74i)10-s + (5.81 + 10.0i)11-s + (2.64 + 1.60i)12-s + 4.92i·13-s + (4.37 − 2.52i)14-s + (0.140 − 6.07i)15-s − 10.8·16-s + (1.96 − 3.40i)17-s + ⋯ |
| L(s) = 1 | + 0.861i·2-s + (0.854 + 0.519i)3-s + 0.257·4-s + (−0.202 − 0.350i)5-s + (−0.447 + 0.735i)6-s + (−0.209 − 0.362i)7-s + 1.08i·8-s + (0.459 + 0.888i)9-s + (0.302 − 0.174i)10-s + (0.528 + 0.915i)11-s + (0.220 + 0.134i)12-s + 0.378i·13-s + (0.312 − 0.180i)14-s + (0.00938 − 0.404i)15-s − 0.675·16-s + (0.115 − 0.200i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.127 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.127 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.39175 + 1.58233i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39175 + 1.58233i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.56 - 1.55i)T \) |
| 19 | \( 1 + (-1.86 + 18.9i)T \) |
| good | 2 | \( 1 - 1.72iT - 4T^{2} \) |
| 5 | \( 1 + (1.01 + 1.75i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (1.46 + 2.53i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-5.81 - 10.0i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 4.92iT - 169T^{2} \) |
| 17 | \( 1 + (-1.96 + 3.40i)T + (-144.5 - 250. i)T^{2} \) |
| 23 | \( 1 + 3.89T + 529T^{2} \) |
| 29 | \( 1 + (4.35 + 2.51i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (21.7 + 12.5i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 60.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (22.1 - 12.7i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + 0.0714T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-26.9 + 46.6i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (48.4 - 27.9i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-95.8 + 55.3i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (9.52 - 16.4i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 - 56.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (14.4 + 8.34i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-25.6 + 44.5i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 - 45.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (7.15 + 12.3i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (31.2 - 18.0i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 107. iT - 9.40e3T^{2} \) |
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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
−13.02251740966707220568274999474, −11.80256110042059823052793450807, −10.67219818725071139762122158581, −9.498333626356804242779270469888, −8.640294622052215928227280087805, −7.51258774897353666884317483887, −6.79324928887470954406095407457, −5.14920621411855178932684065913, −4.01186231318133839586295572210, −2.26224740661278167182544498010,
1.41349222680416951085380135444, 2.93959739907057355597611382261, 3.68628133949065959146737859392, 6.03869608516782924610362608659, 7.07618182399291455753060722061, 8.225271372610804426073217911848, 9.309241287967004708201005788416, 10.33976761206585020012565476326, 11.39629441557051671551777824657, 12.25254311052141363854651289408
