| L(s) = 1 | + (−0.711 + 1.86i)2-s + (−2.98 − 2.66i)4-s − 4.97·5-s + 12.2i·7-s + (7.09 − 3.68i)8-s + (3.54 − 9.30i)10-s + 13.4i·11-s + 14.1·13-s + (−22.9 − 8.72i)14-s + (1.84 + 15.8i)16-s + 5.89·17-s + 4.35i·19-s + (14.8 + 13.2i)20-s + (−25.1 − 9.57i)22-s − 0.906i·23-s + ⋯ |
| L(s) = 1 | + (−0.355 + 0.934i)2-s + (−0.746 − 0.665i)4-s − 0.995·5-s + 1.75i·7-s + (0.887 − 0.461i)8-s + (0.354 − 0.930i)10-s + 1.22i·11-s + 1.09·13-s + (−1.63 − 0.623i)14-s + (0.115 + 0.993i)16-s + 0.346·17-s + 0.229i·19-s + (0.743 + 0.662i)20-s + (−1.14 − 0.435i)22-s − 0.0394i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.665 + 0.746i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6372765134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6372765134\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.711 - 1.86i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
| good | 5 | \( 1 + 4.97T + 25T^{2} \) |
| 7 | \( 1 - 12.2iT - 49T^{2} \) |
| 11 | \( 1 - 13.4iT - 121T^{2} \) |
| 13 | \( 1 - 14.1T + 169T^{2} \) |
| 17 | \( 1 - 5.89T + 289T^{2} \) |
| 23 | \( 1 + 0.906iT - 529T^{2} \) |
| 29 | \( 1 - 10.3T + 841T^{2} \) |
| 31 | \( 1 - 43.2iT - 961T^{2} \) |
| 37 | \( 1 + 1.61T + 1.36e3T^{2} \) |
| 41 | \( 1 + 69.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + 32.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 38.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 8.31T + 2.80e3T^{2} \) |
| 59 | \( 1 + 20.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 118.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 57.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 11.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 23.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 0.286iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 24.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 43.7T + 7.92e3T^{2} \) |
| 97 | \( 1 - 115.T + 9.40e3T^{2} \) |
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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
−10.65189337941285363334811771264, −9.725517618539229733058180343367, −8.731119608155811650237851152338, −8.389600572724155250076198836338, −7.37715307684658278180825614255, −6.47132975572537259171620526965, −5.52105548052745348666244375902, −4.68315024039399351182036432756, −3.44728250232358340823856583552, −1.73479722321722610310385834901,
0.30224066522737491407661488163, 1.21769723287882998510702962190, 3.26262019375012226970331676750, 3.78584347269441346206524519297, 4.62691892826933773137912002358, 6.24236159710112509452645887354, 7.52918129781342131689551920519, 7.994563219284779894445945750763, 8.854548751408863094897181263453, 9.986573464650771195950134376365
