| L(s) = 1 | + (0.986 + 1.42i)3-s + (2.52 + 0.917i)5-s + (−0.168 + 0.957i)7-s + (−1.05 + 2.80i)9-s + (−0.297 + 0.108i)11-s + (−1.15 + 0.973i)13-s + (1.17 + 4.49i)15-s + (−0.587 − 1.01i)17-s + (3.11 − 5.38i)19-s + (−1.52 + 0.703i)21-s + (−0.375 − 2.12i)23-s + (1.68 + 1.41i)25-s + (−5.03 + 1.26i)27-s + (−3.37 − 2.83i)29-s + (1.50 + 8.54i)31-s + ⋯ |
| L(s) = 1 | + (0.569 + 0.822i)3-s + (1.12 + 0.410i)5-s + (−0.0638 + 0.361i)7-s + (−0.351 + 0.936i)9-s + (−0.0897 + 0.0326i)11-s + (−0.321 + 0.269i)13-s + (0.304 + 1.16i)15-s + (−0.142 − 0.246i)17-s + (0.713 − 1.23i)19-s + (−0.333 + 0.153i)21-s + (−0.0783 − 0.444i)23-s + (0.336 + 0.282i)25-s + (−0.969 + 0.243i)27-s + (−0.626 − 0.525i)29-s + (0.270 + 1.53i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.54795 + 1.09975i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.54795 + 1.09975i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.986 - 1.42i)T \) |
| good | 5 | \( 1 + (-2.52 - 0.917i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.168 - 0.957i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (0.297 - 0.108i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (1.15 - 0.973i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.587 + 1.01i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.11 + 5.38i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.375 + 2.12i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (3.37 + 2.83i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.50 - 8.54i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-2.23 - 3.86i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.47 + 3.75i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-5.25 + 1.91i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.429 + 2.43i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + (1.62 + 0.589i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.176 + 0.999i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (0.656 - 0.550i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (4.79 + 8.31i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.62 + 13.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.59 - 7.20i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (3.58 + 3.01i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-7.74 + 13.4i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.21 + 1.89i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.02995419245968645396979397720, −10.33244087622311890149239821059, −9.397827437323368939114690487869, −9.040623699051757969953974029663, −7.73116643662798947034771754452, −6.58666683700064406288271388285, −5.48571681383217704497362709515, −4.59757723898915066802903903236, −3.06886955640931298277789236553, −2.20279525681424345109437818477,
1.30592209775080635955812140695, 2.47671162099533558754071985177, 3.87846408354110806073806410102, 5.52235370082492066549076342038, 6.17699041013390527710192381507, 7.43307715701964364881289367338, 8.089630999690942501294052652937, 9.372149984504191488360569182919, 9.709860295136885666006968590309, 10.96266331350135073248605747628
