L(s) = 1 | + (−0.707 − 1.22i)2-s + (−0.999 + 1.73i)4-s + (−1.24 + 4.84i)5-s + (0.315 − 0.182i)7-s + 2.82·8-s + (6.81 − 1.89i)10-s + (8.32 − 4.80i)11-s + (−6.62 − 3.82i)13-s + (−0.446 − 0.257i)14-s + (−2.00 − 3.46i)16-s + 12.3·17-s + 5.36·19-s + (−7.13 − 7.00i)20-s + (−11.7 − 6.79i)22-s + (−6.03 + 10.4i)23-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.249 + 0.968i)5-s + (0.0451 − 0.0260i)7-s + 0.353·8-s + (0.681 − 0.189i)10-s + (0.756 − 0.436i)11-s + (−0.509 − 0.294i)13-s + (−0.0319 − 0.0184i)14-s + (−0.125 − 0.216i)16-s + 0.725·17-s + 0.282·19-s + (−0.356 − 0.350i)20-s + (−0.535 − 0.308i)22-s + (−0.262 + 0.454i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.289636753\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.289636753\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 1.22i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.24 - 4.84i)T \) |
good | 7 | \( 1 + (-0.315 + 0.182i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-8.32 + 4.80i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (6.62 + 3.82i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 12.3T + 289T^{2} \) |
| 19 | \( 1 - 5.36T + 361T^{2} \) |
| 23 | \( 1 + (6.03 - 10.4i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-27.1 + 15.6i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (2.45 - 4.26i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 40.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-54.8 - 31.6i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (46.6 - 26.9i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-14.1 - 24.4i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 41.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-97.2 - 56.1i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (22.9 + 39.8i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-39.9 - 23.0i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 125. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 59.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (25.3 + 43.8i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-29.5 - 51.1i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 8.93iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-113. + 65.6i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−10.00601512337744160952242017685, −9.674818276117308671532984078447, −8.394253879367630368641400502528, −7.72671452884530912016839908217, −6.79059603273110541603597098822, −5.85534948691353840813618267423, −4.48925562014915372576325598555, −3.42325863725296290377144915966, −2.65718696449655002720252318637, −1.12920490706244648766063540839,
0.58846200733799633158560967240, 1.89239746547901175036765595109, 3.73479170633874555460279345426, 4.71897294990144713443919492206, 5.48769337035591823919787504760, 6.59433964615135858015688256806, 7.45412838641469936935276611024, 8.263113258410650865494728380254, 9.082861751183256945049989723196, 9.662852283441468930300074836305