| L(s) = 1 | + (0.635 − 1.89i)2-s + (−3.19 − 2.40i)4-s − 10.1·7-s + (−6.59 + 4.52i)8-s − 10.6i·11-s + 11.7i·13-s + (−6.41 + 19.1i)14-s + (4.38 + 15.3i)16-s + 16.6i·17-s + 0.464i·19-s + (−20.2 − 6.77i)22-s + 42.1·23-s + (22.3 + 7.47i)26-s + (32.2 + 24.3i)28-s − 19.5·29-s + ⋯ |
| L(s) = 1 | + (0.317 − 0.948i)2-s + (−0.798 − 0.602i)4-s − 1.44·7-s + (−0.824 + 0.565i)8-s − 0.968i·11-s + 0.905i·13-s + (−0.458 + 1.36i)14-s + (0.274 + 0.961i)16-s + 0.978i·17-s + 0.0244i·19-s + (−0.918 − 0.307i)22-s + 1.83·23-s + (0.858 + 0.287i)26-s + (1.15 + 0.869i)28-s − 0.674·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.181i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.983 + 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.207953630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.207953630\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.635 + 1.89i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 10.1T + 49T^{2} \) |
| 11 | \( 1 + 10.6iT - 121T^{2} \) |
| 13 | \( 1 - 11.7iT - 169T^{2} \) |
| 17 | \( 1 - 16.6iT - 289T^{2} \) |
| 19 | \( 1 - 0.464iT - 361T^{2} \) |
| 23 | \( 1 - 42.1T + 529T^{2} \) |
| 29 | \( 1 + 19.5T + 841T^{2} \) |
| 31 | \( 1 + 9.17iT - 961T^{2} \) |
| 37 | \( 1 + 23.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 44.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 58.3T + 1.84e3T^{2} \) |
| 47 | \( 1 - 41.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 80.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 63.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 80.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 22.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 61.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 137. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 138. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 86.2T + 6.88e3T^{2} \) |
| 89 | \( 1 - 127.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 15iT - 9.40e3T^{2} \) |
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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
−9.990266009081227716137877164073, −9.090102250748707413633964313778, −8.739424552874356641825870054148, −7.18935622983923421789883749153, −6.21385322724770095201079099316, −5.53931527309328591136840452568, −4.17109242921606507179495505081, −3.43901978886761551095168553516, −2.50464916781923126231265647483, −0.967173613985899420153738161496,
0.45201746814997167362310853198, 2.80504514691723490474996260075, 3.58709444372929285408263800267, 4.85705888812678003015000796605, 5.56219295993666833845847337299, 6.76780527266300700690441186070, 7.04341876825920466323019598625, 8.085087250106160876284298463311, 9.229265162744708996753071772807, 9.587133795766934163364636706597
