| L(s) = 1 | + 1.41i·2-s + (−1 − 2.82i)3-s − 2.00·4-s − 8.48i·5-s + (4.00 − 1.41i)6-s − 2.82i·8-s + (−7.00 + 5.65i)9-s + 12·10-s + (2.00 + 5.65i)12-s − 13-s + (−24 + 8.48i)15-s + 4.00·16-s + 8.48i·17-s + (−8.00 − 9.89i)18-s − 31·19-s + 16.9i·20-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (−0.333 − 0.942i)3-s − 0.500·4-s − 1.69i·5-s + (0.666 − 0.235i)6-s − 0.353i·8-s + (−0.777 + 0.628i)9-s + 1.20·10-s + (0.166 + 0.471i)12-s − 0.0769·13-s + (−1.60 + 0.565i)15-s + 0.250·16-s + 0.499i·17-s + (−0.444 − 0.549i)18-s − 1.63·19-s + 0.848i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 + 0.333i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.942 + 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.102363 - 0.596619i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.102363 - 0.596619i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + (1 + 2.82i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 8.48iT - 25T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + T + 169T^{2} \) |
| 17 | \( 1 - 8.48iT - 289T^{2} \) |
| 19 | \( 1 + 31T + 361T^{2} \) |
| 23 | \( 1 - 8.48iT - 529T^{2} \) |
| 29 | \( 1 + 16.9iT - 841T^{2} \) |
| 31 | \( 1 + 7T + 961T^{2} \) |
| 37 | \( 1 + T + 1.36e3T^{2} \) |
| 41 | \( 1 + 33.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 31T + 1.84e3T^{2} \) |
| 47 | \( 1 + 42.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 25.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 8.48iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 50T + 3.72e3T^{2} \) |
| 67 | \( 1 - 65T + 4.48e3T^{2} \) |
| 71 | \( 1 - 59.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 97T + 5.32e3T^{2} \) |
| 79 | \( 1 + 103T + 6.24e3T^{2} \) |
| 83 | \( 1 + 42.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 118. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 166T + 9.40e3T^{2} \) |
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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
−11.42246508583110982931956675307, −10.02675497105011586656715882800, −8.639924403211569938747714019135, −8.433890274118731173033881026709, −7.23967137742918327717320971194, −6.09308076735885301058227264457, −5.27471474430816613011688006242, −4.20952292335642111774566553367, −1.78804504103841022946141001341, −0.30063623931840240504964648974,
2.50449225655717400743056059183, 3.47280463764752374706074937287, 4.57936349551115430058826044095, 6.00126048529382743648643433039, 6.92873019094949383832012862066, 8.390217434238413612560123698455, 9.559721191266592519689023540103, 10.35917386067993420639895914539, 10.92546721175649204149241301385, 11.54099593810411389838942966422
