| L(s) = 1 | − 2.86·5-s + (2.43 − 1.03i)7-s − 3.32i·11-s + 0.821i·13-s − 3.52·17-s + 5.25i·19-s − 5.12i·23-s + 3.22·25-s + 1.64i·29-s − 2.55i·31-s + (−6.98 + 2.96i)35-s − 8.93·37-s − 3.49·41-s − 0.161·43-s − 5.34·47-s + ⋯ |
| L(s) = 1 | − 1.28·5-s + (0.920 − 0.391i)7-s − 1.00i·11-s + 0.227i·13-s − 0.853·17-s + 1.20i·19-s − 1.06i·23-s + 0.645·25-s + 0.305i·29-s − 0.458i·31-s + (−1.18 + 0.501i)35-s − 1.46·37-s − 0.546·41-s − 0.0245·43-s − 0.779·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4408086705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4408086705\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.43 + 1.03i)T \) |
| good | 5 | \( 1 + 2.86T + 5T^{2} \) |
| 11 | \( 1 + 3.32iT - 11T^{2} \) |
| 13 | \( 1 - 0.821iT - 13T^{2} \) |
| 17 | \( 1 + 3.52T + 17T^{2} \) |
| 19 | \( 1 - 5.25iT - 19T^{2} \) |
| 23 | \( 1 + 5.12iT - 23T^{2} \) |
| 29 | \( 1 - 1.64iT - 29T^{2} \) |
| 31 | \( 1 + 2.55iT - 31T^{2} \) |
| 37 | \( 1 + 8.93T + 37T^{2} \) |
| 41 | \( 1 + 3.49T + 41T^{2} \) |
| 43 | \( 1 + 0.161T + 43T^{2} \) |
| 47 | \( 1 + 5.34T + 47T^{2} \) |
| 53 | \( 1 - 3.28iT - 53T^{2} \) |
| 59 | \( 1 - 4.08T + 59T^{2} \) |
| 61 | \( 1 + 8.43iT - 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 + 5.68iT - 71T^{2} \) |
| 73 | \( 1 + 7.86iT - 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 + 12.8iT - 97T^{2} \) |
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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
−8.671673458000920641228861037083, −8.435607083853198030345712481535, −7.62749910050109477122261498834, −6.85645308870061996632319721735, −5.82999965854793701977399674206, −4.71600919288447445724510812435, −4.05839441107708363829868520640, −3.20021175977962821152961646515, −1.68158952878483157836837160605, −0.17392485673914249283525343340,
1.63566004374256433867032824141, 2.85799605604592281019898958359, 4.06483046508916499175931824892, 4.70612254876644272740305796044, 5.50858902812228940299980027150, 6.99517706788968221674512662392, 7.29506347752379461651844279509, 8.316494109763063886168657657628, 8.769326256828283739155675317165, 9.793320315997389442833584992427
