| L(s) = 1 | + (0.656 − 1.88i)2-s + (−3.13 − 2.48i)4-s − 0.894·5-s − 1.58i·7-s + (−6.74 + 4.29i)8-s + (−0.587 + 1.69i)10-s − 12.3i·11-s − 11.5·13-s + (−2.98 − 1.03i)14-s + (3.68 + 15.5i)16-s − 26.0·17-s + 21.4i·19-s + (2.80 + 2.22i)20-s + (−23.3 − 8.12i)22-s − 27.4i·23-s + ⋯ |
| L(s) = 1 | + (0.328 − 0.944i)2-s + (−0.784 − 0.620i)4-s − 0.178·5-s − 0.225i·7-s + (−0.843 + 0.537i)8-s + (−0.0587 + 0.169i)10-s − 1.12i·11-s − 0.888·13-s + (−0.213 − 0.0741i)14-s + (0.230 + 0.973i)16-s − 1.53·17-s + 1.12i·19-s + (0.140 + 0.111i)20-s + (−1.06 − 0.369i)22-s − 1.19i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.203986 + 0.586791i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.203986 + 0.586791i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.656 + 1.88i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 0.894T + 25T^{2} \) |
| 7 | \( 1 + 1.58iT - 49T^{2} \) |
| 11 | \( 1 + 12.3iT - 121T^{2} \) |
| 13 | \( 1 + 11.5T + 169T^{2} \) |
| 17 | \( 1 + 26.0T + 289T^{2} \) |
| 19 | \( 1 - 21.4iT - 361T^{2} \) |
| 23 | \( 1 + 27.4iT - 529T^{2} \) |
| 29 | \( 1 - 9.49T + 841T^{2} \) |
| 31 | \( 1 - 49.1iT - 961T^{2} \) |
| 37 | \( 1 + 20.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 5.25T + 1.68e3T^{2} \) |
| 43 | \( 1 + 80.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 60.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 12.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + 30.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 56.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 70.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 71.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 31.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 96.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 24.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 68.1T + 7.92e3T^{2} \) |
| 97 | \( 1 - 100.T + 9.40e3T^{2} \) |
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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
−10.75084067136415033055252245505, −10.28628931892229119445435903189, −8.997881769554783999627109006107, −8.319001282015180521535372947012, −6.80278698517935745072998492833, −5.63042741022282572822968609147, −4.49190541269092667675327777006, −3.43506673677200779047436209895, −2.09407421549680295836108363215, −0.24954303309184439866212039819,
2.49534427979192537425559167714, 4.20306811086385230895426254230, 4.93082604839774756340402680562, 6.20327663278499287235130553534, 7.19572712694385303461955234072, 7.86430450810467940126790560387, 9.197635420035207713105585178096, 9.665922738766389284274789584499, 11.23404765956271755926703570997, 12.11102472947818937460260412006
