| L(s) = 1 | + (−1.65 + 4.54i)5-s + (1.68 − 9.56i)7-s + (−5.23 − 14.3i)11-s + (−6.28 + 5.27i)13-s + (9.01 − 5.20i)17-s + (7.17 − 12.4i)19-s + (24.2 − 4.27i)23-s + (1.22 + 1.02i)25-s + (20.0 − 23.8i)29-s + (−10.5 − 59.6i)31-s + (40.6 + 23.4i)35-s + (−0.367 − 0.636i)37-s + (−30.1 − 35.9i)41-s + (−69.9 + 25.4i)43-s + (12.5 + 2.21i)47-s + ⋯ |
| L(s) = 1 | + (−0.330 + 0.909i)5-s + (0.240 − 1.36i)7-s + (−0.476 − 1.30i)11-s + (−0.483 + 0.405i)13-s + (0.530 − 0.306i)17-s + (0.377 − 0.654i)19-s + (1.05 − 0.185i)23-s + (0.0489 + 0.0410i)25-s + (0.690 − 0.822i)29-s + (−0.339 − 1.92i)31-s + (1.16 + 0.671i)35-s + (−0.00992 − 0.0171i)37-s + (−0.735 − 0.876i)41-s + (−1.62 + 0.592i)43-s + (0.266 + 0.0470i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.267 + 0.963i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.267 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.03702 - 0.788568i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03702 - 0.788568i\) |
| \(L(2)\) |
|
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 | \( 1 + (1.65 - 4.54i)T + (-19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (-1.68 + 9.56i)T + (-46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (5.23 + 14.3i)T + (-92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (6.28 - 5.27i)T + (29.3 - 166. i)T^{2} \) |
| 17 | \( 1 + (-9.01 + 5.20i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-7.17 + 12.4i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-24.2 + 4.27i)T + (497. - 180. i)T^{2} \) |
| 29 | \( 1 + (-20.0 + 23.8i)T + (-146. - 828. i)T^{2} \) |
| 31 | \( 1 + (10.5 + 59.6i)T + (-903. + 328. i)T^{2} \) |
| 37 | \( 1 + (0.367 + 0.636i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (30.1 + 35.9i)T + (-291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (69.9 - 25.4i)T + (1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-12.5 - 2.21i)T + (2.07e3 + 755. i)T^{2} \) |
| 53 | \( 1 + 36.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (30.5 - 83.8i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (7.06 - 40.0i)T + (-3.49e3 - 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-61.6 + 51.6i)T + (779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (0.595 - 0.343i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-13.7 + 23.8i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-97.1 - 81.5i)T + (1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-8.62 + 10.2i)T + (-1.19e3 - 6.78e3i)T^{2} \) |
| 89 | \( 1 + (-146. - 84.4i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (60.2 - 21.9i)T + (7.20e3 - 6.04e3i)T^{2} \) |
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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
−11.08533905299898953496672922241, −10.51578767028162070889633436993, −9.478518312780941089955461736683, −8.120143213721042987677320949143, −7.34529375552025534554408171112, −6.55795439410894013907874813211, −5.13253714610824491372851979894, −3.84423503339660414405234716387, −2.82131811978219696431967678740, −0.64129212646749262132679454013,
1.62747429162477325620733019012, 3.11714162253895111470080692775, 4.99012472831360148485937457782, 5.17757240755559970455944571160, 6.81165480755583839475385242471, 8.000764844888438684629678681742, 8.697303068541570895097870172346, 9.631176173243362443232306221442, 10.58457061937158231833124262464, 12.05872375195176456033464389782
