L(s) = 1 | + (0.419 + 2.97i)3-s + (−8.20 − 4.73i)5-s + (−1.05 − 1.83i)7-s + (−8.64 + 2.49i)9-s + (−13.7 + 7.91i)11-s + (4.70 − 8.14i)13-s + (10.6 − 26.3i)15-s − 11.6i·17-s − 12.9·19-s + (4.99 − 3.90i)21-s + (5.27 + 3.04i)23-s + (32.4 + 56.1i)25-s + (−11.0 − 24.6i)27-s + (−24.7 + 14.2i)29-s + (−8.75 + 15.1i)31-s + ⋯ |
L(s) = 1 | + (0.139 + 0.990i)3-s + (−1.64 − 0.947i)5-s + (−0.150 − 0.261i)7-s + (−0.960 + 0.276i)9-s + (−1.24 + 0.719i)11-s + (0.361 − 0.626i)13-s + (0.708 − 1.75i)15-s − 0.682i·17-s − 0.682·19-s + (0.237 − 0.186i)21-s + (0.229 + 0.132i)23-s + (1.29 + 2.24i)25-s + (−0.408 − 0.912i)27-s + (−0.854 + 0.493i)29-s + (−0.282 + 0.489i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 + 0.439i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.898 + 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00767223 - 0.0331310i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00767223 - 0.0331310i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.419 - 2.97i)T \) |
good | 5 | \( 1 + (8.20 + 4.73i)T + (12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (1.05 + 1.83i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (13.7 - 7.91i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-4.70 + 8.14i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 11.6iT - 289T^{2} \) |
| 19 | \( 1 + 12.9T + 361T^{2} \) |
| 23 | \( 1 + (-5.27 - 3.04i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (24.7 - 14.2i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (8.75 - 15.1i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 15.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-14.8 - 8.54i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-21.7 - 37.6i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (20.6 - 11.9i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 14.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (38.5 + 22.2i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (1.86 + 3.22i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (21.0 - 36.3i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 120. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.48T + 5.32e3T^{2} \) |
| 79 | \( 1 + (60.5 + 104. i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (46.5 - 26.8i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 102. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (58.9 + 102. i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−12.43487970577712356461064889480, −11.29197469027511974714322829136, −10.50862576661076132350179104839, −9.225477477908854952163289689266, −8.239847199079663694699549719078, −7.44928055078729426842713606696, −5.25758312208750005943862217554, −4.44308875797039482201019158037, −3.26547724910180908429594352918, −0.02082722439602907141299993727,
2.63823956140511072312027052039, 3.87673321293001182425882626827, 5.93634907568890801650644603800, 7.08033796819116596575733172657, 7.918394544196283837294000447402, 8.688695857375195106935756573450, 10.70661166423332070062437340936, 11.29427574471257982832403504227, 12.26628065899441645717686806209, 13.16412197044152952987730156324