| L(s) = 1 | + (−0.409 + 1.68i)3-s + (−0.103 + 0.584i)5-s + (−2.18 + 1.83i)7-s + (−2.66 − 1.37i)9-s + (−0.0708 − 0.402i)11-s + (−0.182 + 0.0664i)13-s + (−0.942 − 0.413i)15-s + (−3.66 + 6.34i)17-s + (−2.06 − 3.57i)19-s + (−2.19 − 4.43i)21-s + (−3.12 − 2.61i)23-s + (4.36 + 1.58i)25-s + (3.41 − 3.92i)27-s + (−9.66 − 3.51i)29-s + (4.78 + 4.01i)31-s + ⋯ |
| L(s) = 1 | + (−0.236 + 0.971i)3-s + (−0.0461 + 0.261i)5-s + (−0.826 + 0.693i)7-s + (−0.888 − 0.459i)9-s + (−0.0213 − 0.121i)11-s + (−0.0506 + 0.0184i)13-s + (−0.243 − 0.106i)15-s + (−0.888 + 1.53i)17-s + (−0.473 − 0.820i)19-s + (−0.478 − 0.967i)21-s + (−0.650 − 0.546i)23-s + (0.873 + 0.317i)25-s + (0.656 − 0.754i)27-s + (−1.79 − 0.653i)29-s + (0.858 + 0.720i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.133i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0421726 + 0.627270i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0421726 + 0.627270i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.409 - 1.68i)T \) |
| good | 5 | \( 1 + (0.103 - 0.584i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (2.18 - 1.83i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (0.0708 + 0.402i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (0.182 - 0.0664i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (3.66 - 6.34i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.06 + 3.57i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.12 + 2.61i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (9.66 + 3.51i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-4.78 - 4.01i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (2.88 - 4.99i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.92 + 2.88i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.03 - 5.89i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (8.62 - 7.23i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 3.42T + 53T^{2} \) |
| 59 | \( 1 + (0.813 - 4.61i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-3.34 + 2.80i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-11.4 + 4.15i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (4.09 - 7.09i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.96 - 3.41i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.47 - 3.08i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-7.91 - 2.88i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-2.38 - 4.13i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.34 - 7.62i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.27797738901185423483268950321, −10.75852144445656635667836737122, −9.762410099574329109581569747433, −9.031864227554280299022427157804, −8.211417209160504629821147761720, −6.55607136788315671269561585416, −6.02778163976530427402230288365, −4.75649505030772481317780536492, −3.69885639829323074212421958694, −2.54595552767272275755421964195,
0.38632646496038588364744274762, 2.15687573205413364132162153793, 3.61158861999138512482232314264, 5.01839393893977980704068454056, 6.17781563496850032971736903418, 7.02048684328474436105496197318, 7.72302618100108460594999911357, 8.885156456987108419978887574839, 9.788923459777712593616271834875, 10.87716068497551217166973565797
