| L(s) = 1 | + (2.19 + 0.191i)2-s + (2.79 + 0.492i)4-s + (2.15 − 0.605i)5-s + (0.000530 + 0.000371i)7-s + (1.77 + 0.474i)8-s + (4.83 − 0.913i)10-s + (−0.925 + 2.54i)11-s + (−0.277 − 3.16i)13-s + (0.00109 + 0.000915i)14-s + (−1.53 − 0.559i)16-s + (−6.58 + 1.76i)17-s + (1.52 − 0.879i)19-s + (6.30 − 0.630i)20-s + (−2.51 + 5.39i)22-s + (2.93 + 4.18i)23-s + ⋯ |
| L(s) = 1 | + (1.54 + 0.135i)2-s + (1.39 + 0.246i)4-s + (0.962 − 0.270i)5-s + (0.000200 + 0.000140i)7-s + (0.626 + 0.167i)8-s + (1.52 − 0.288i)10-s + (−0.278 + 0.766i)11-s + (−0.0768 − 0.878i)13-s + (0.000291 + 0.000244i)14-s + (−0.384 − 0.139i)16-s + (−1.59 + 0.428i)17-s + (0.349 − 0.201i)19-s + (1.41 − 0.140i)20-s + (−0.535 + 1.14i)22-s + (0.611 + 0.872i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.28672 + 0.142431i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.28672 + 0.142431i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.15 + 0.605i)T \) |
| good | 2 | \( 1 + (-2.19 - 0.191i)T + (1.96 + 0.347i)T^{2} \) |
| 7 | \( 1 + (-0.000530 - 0.000371i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (0.925 - 2.54i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.277 + 3.16i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (6.58 - 1.76i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.52 + 0.879i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.93 - 4.18i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-4.37 + 3.67i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.944 - 5.35i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (0.0678 + 0.253i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (7.56 - 9.01i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (3.67 + 7.88i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-0.403 + 0.576i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (5.73 - 5.73i)T - 53iT^{2} \) |
| 59 | \( 1 + (-2.11 + 0.768i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (1.14 + 6.48i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.47 + 0.216i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-10.4 - 6.02i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.375 - 1.40i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-3.36 - 4.00i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.39 + 15.9i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (2.99 + 5.18i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.62 + 3.55i)T + (62.3 - 74.3i)T^{2} \) |
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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
−11.54227836173544908097367378528, −10.51160892435910832101100546408, −9.578936797636561610599788110561, −8.484904815744510450673906654417, −7.04133178303660214775178059392, −6.30208753872059043826004775201, −5.21803317072072942147808082360, −4.69523525656989048976328773734, −3.24197308351325018359779405594, −2.07176177011250510352868676677,
2.13785361222183083549442324322, 3.11779132633162273764696059236, 4.47126651186396261173495086745, 5.28876629154028676585226997403, 6.37044141769066460685534326604, 6.85407583497941279694528296483, 8.610604268692923025985463334316, 9.478166656652495397771638905333, 10.78908378127704479239596535713, 11.29478970649071293280634871650
