| L(s) = 1 | + 1.73i·2-s + (−1.5 − 2.59i)3-s + 1.00·4-s + (−3 − 1.73i)5-s + (4.5 − 2.59i)6-s + 8.66i·8-s + (−4.5 + 7.79i)9-s + (2.99 − 5.19i)10-s + (1.5 − 0.866i)11-s + (−1.50 − 2.59i)12-s + (2 + 3.46i)13-s + 10.3i·15-s − 10.9·16-s + (13.5 + 7.79i)17-s + (−13.5 − 7.79i)18-s + (−5.5 − 9.52i)19-s + ⋯ |
| L(s) = 1 | + 0.866i·2-s + (−0.5 − 0.866i)3-s + 0.250·4-s + (−0.600 − 0.346i)5-s + (0.750 − 0.433i)6-s + 1.08i·8-s + (−0.5 + 0.866i)9-s + (0.299 − 0.519i)10-s + (0.136 − 0.0787i)11-s + (−0.125 − 0.216i)12-s + (0.153 + 0.266i)13-s + 0.692i·15-s − 0.687·16-s + (0.794 + 0.458i)17-s + (−0.750 − 0.433i)18-s + (−0.289 − 0.501i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.540 - 0.841i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.28151 + 0.699609i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28151 + 0.699609i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.5 + 2.59i)T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.73iT - 4T^{2} \) |
| 5 | \( 1 + (3 + 1.73i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 0.866i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-2 - 3.46i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + (-13.5 - 7.79i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (5.5 + 9.52i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-24 - 13.8i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-39 - 22.5i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 - 32T + 961T^{2} \) |
| 37 | \( 1 + (-17 - 29.4i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (10.5 - 6.06i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-30.5 + 52.8i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 - 48.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 - 50.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 56T + 3.72e3T^{2} \) |
| 67 | \( 1 + 31T + 4.48e3T^{2} \) |
| 71 | \( 1 - 31.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (32.5 - 56.2i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 - 38T + 6.24e3T^{2} \) |
| 83 | \( 1 + (42 + 24.2i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (108 - 62.3i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-57.5 + 99.5i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−11.30704143673092850690197596530, −10.34137919385693955204457431290, −8.714025859843364598546306762638, −8.116718544770657019058916020839, −7.22953465595280053776904021674, −6.52590256006308433673962941432, −5.61831408657135468014299477922, −4.60091723265448127653998836105, −2.78506120359528270258312601508, −1.16713103074793265534927428255,
0.791861851556797233383498870812, 2.77498822113853578336385968400, 3.64508853238292063849600193329, 4.65206702712917515910319057020, 5.99835311765873498431896863198, 6.93678753518574773483949803314, 8.115565213032866939296626031528, 9.383088613660855499231476493111, 10.16994052149051360475148903293, 10.79099681125557011835123606332
