| L(s) = 1 | − 2-s + 4-s + (0.648 + 2.56i)7-s − 8-s − 1.29i·11-s − 3.13·13-s + (−0.648 − 2.56i)14-s + 16-s − 5.53i·17-s − 7.37i·19-s + 1.29i·22-s − 1.83·23-s + 3.13·26-s + (0.648 + 2.56i)28-s − 1.83i·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + (0.245 + 0.969i)7-s − 0.353·8-s − 0.390i·11-s − 0.868·13-s + (−0.173 − 0.685i)14-s + 0.250·16-s − 1.34i·17-s − 1.69i·19-s + 0.276i·22-s − 0.382·23-s + 0.613·26-s + (0.122 + 0.484i)28-s − 0.340i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2022494175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2022494175\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.648 - 2.56i)T \) |
| good | 11 | \( 1 + 1.29iT - 11T^{2} \) |
| 13 | \( 1 + 3.13T + 13T^{2} \) |
| 17 | \( 1 + 5.53iT - 17T^{2} \) |
| 19 | \( 1 + 7.37iT - 19T^{2} \) |
| 23 | \( 1 + 1.83T + 23T^{2} \) |
| 29 | \( 1 + 1.83iT - 29T^{2} \) |
| 31 | \( 1 - 10.4iT - 31T^{2} \) |
| 37 | \( 1 - 10.6iT - 37T^{2} \) |
| 41 | \( 1 - 3.13T + 41T^{2} \) |
| 43 | \( 1 - 3.53iT - 43T^{2} \) |
| 47 | \( 1 - 10.7iT - 47T^{2} \) |
| 53 | \( 1 + 4.42T + 53T^{2} \) |
| 59 | \( 1 + 7.18T + 59T^{2} \) |
| 61 | \( 1 + 4.88iT - 61T^{2} \) |
| 67 | \( 1 - 9.79iT - 67T^{2} \) |
| 71 | \( 1 + 7.37iT - 71T^{2} \) |
| 73 | \( 1 + 3.40T + 73T^{2} \) |
| 79 | \( 1 + 9.01T + 79T^{2} \) |
| 83 | \( 1 + 6.26iT - 83T^{2} \) |
| 89 | \( 1 - 7.94T + 89T^{2} \) |
| 97 | \( 1 + 8.09T + 97T^{2} \) |
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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
−9.170746385507936200932940365273, −8.358777577143606505165251620682, −7.63588246164962866650905289321, −6.86576591277872111764710143696, −6.18561848249280045863856634852, −5.08918244875897183488540201689, −4.70309168254699696766387623113, −2.92090148488734252992407009456, −2.70130752775983548464646715240, −1.33062058394652764360774399003,
0.079188706485313008974897166107, 1.50932956047106054712998559153, 2.27448640888268070174712476174, 3.74756954378743711643184366801, 4.17151226634443720677987034365, 5.44485915029638928470307586576, 6.15769458811971019022711453216, 7.07449549057329479235050948232, 7.76870417712743952015042485688, 8.096511392413405717724126341366
