L(s) = 1 | − i·2-s − 4-s + (−1.60 + 2.77i)5-s + (0.817 − 2.51i)7-s + i·8-s + (2.77 + 1.60i)10-s + (−3.80 − 2.19i)11-s + (1.68 + 3.19i)13-s + (−2.51 − 0.817i)14-s + 16-s + 1.88·17-s + (−0.507 + 0.292i)19-s + (1.60 − 2.77i)20-s + (−2.19 + 3.80i)22-s − 4.42i·23-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (−0.716 + 1.24i)5-s + (0.309 − 0.951i)7-s + 0.353i·8-s + (0.877 + 0.506i)10-s + (−1.14 − 0.663i)11-s + (0.466 + 0.884i)13-s + (−0.672 − 0.218i)14-s + 0.250·16-s + 0.456·17-s + (−0.116 + 0.0671i)19-s + (0.358 − 0.620i)20-s + (−0.468 + 0.812i)22-s − 0.922i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.194854809\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.194854809\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.817 + 2.51i)T \) |
| 13 | \( 1 + (-1.68 - 3.19i)T \) |
good | 5 | \( 1 + (1.60 - 2.77i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (3.80 + 2.19i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 - 1.88T + 17T^{2} \) |
| 19 | \( 1 + (0.507 - 0.292i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 4.42iT - 23T^{2} \) |
| 29 | \( 1 + (2.61 - 1.51i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-5.58 + 3.22i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 5.37T + 37T^{2} \) |
| 41 | \( 1 + (-2.62 - 4.54i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.11 + 7.12i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.908 + 1.57i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.48 + 0.858i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 1.92T + 59T^{2} \) |
| 61 | \( 1 + (1.62 - 0.937i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.01 + 12.1i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.32 + 1.91i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-12.9 + 7.50i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.244 + 0.424i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 9.42T + 83T^{2} \) |
| 89 | \( 1 - 9.26T + 89T^{2} \) |
| 97 | \( 1 + (1.90 + 1.10i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.388838243424528426313205859947, −8.220460727833543691340911380669, −7.77911542900765088990565889990, −6.90008327512533778708846657571, −6.05289846747729438228032068402, −4.77625891879982979097379168472, −3.92655418783093407292149988246, −3.20964281143476800360364397789, −2.24482913348074780307621268328, −0.59707562296105750434157837194,
0.986107716644587317317026078199, 2.58178316913848184762044434911, 3.88233059280707409553315089455, 4.91329667641867611347110383568, 5.32374130520523106938213169357, 6.12523140186655350567418258207, 7.54920222996340974859871043621, 7.940691124274956339012673884726, 8.538413298537765175846972470587, 9.310576234909271315818238772219