L(s) = 1 | + (−4.03 + 6.98i)5-s + (3.90 − 2.25i)7-s + (−3.25 + 1.88i)11-s + (3.52 − 6.10i)13-s − 0.517·17-s + 16.4i·19-s + (27.7 + 15.9i)23-s + (−19.9 − 34.6i)25-s + (9.48 + 16.4i)29-s + (13.1 + 7.58i)31-s + 36.3i·35-s − 0.592·37-s + (−12.3 + 21.4i)41-s + (27.8 − 16.0i)43-s + (−52.4 + 30.2i)47-s + ⋯ |
L(s) = 1 | + (−0.806 + 1.39i)5-s + (0.557 − 0.321i)7-s + (−0.296 + 0.171i)11-s + (0.271 − 0.469i)13-s − 0.0304·17-s + 0.864i·19-s + (1.20 + 0.695i)23-s + (−0.799 − 1.38i)25-s + (0.327 + 0.566i)29-s + (0.423 + 0.244i)31-s + 1.03i·35-s − 0.0160·37-s + (−0.301 + 0.522i)41-s + (0.648 − 0.374i)43-s + (−1.11 + 0.644i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 - 0.457i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.889 - 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.105939776\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.105939776\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (4.03 - 6.98i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-3.90 + 2.25i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (3.25 - 1.88i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-3.52 + 6.10i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 0.517T + 289T^{2} \) |
| 19 | \( 1 - 16.4iT - 361T^{2} \) |
| 23 | \( 1 + (-27.7 - 15.9i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-9.48 - 16.4i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-13.1 - 7.58i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 0.592T + 1.36e3T^{2} \) |
| 41 | \( 1 + (12.3 - 21.4i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-27.8 + 16.0i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (52.4 - 30.2i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 0.664T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-30.5 - 17.6i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (33.7 + 58.4i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (74.4 + 42.9i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 56.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 131.T + 5.32e3T^{2} \) |
| 79 | \( 1 + (126. - 73.2i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (87.1 - 50.2i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 25.8T + 7.92e3T^{2} \) |
| 97 | \( 1 + (48.2 + 83.5i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−9.624222665490649274695132722934, −8.411457698087992373212155591524, −7.81501111944822907996948164566, −7.17283579875105111210579109180, −6.46569541953496312001135378943, −5.41026842264473394408655776417, −4.40443866375621955955222926595, −3.43169877962009287425969773052, −2.80541942487674179776300743839, −1.34614047518433391127153022281,
0.32004032463383253297136626806, 1.34792359121168511273762461265, 2.67172175661475191251962993338, 3.96108716334411620251080543555, 4.75767672768839192954298390883, 5.21354992295125114897825738521, 6.39170420216325404028519708100, 7.38462968847199209500628394510, 8.217031207100203211923045569972, 8.726500183988251996775440782653