| L(s) = 1 | + (−0.234 − 0.645i)2-s + (−0.440 − 1.67i)3-s + (1.17 − 0.982i)4-s + (−0.231 − 1.31i)5-s + (−0.977 + 0.677i)6-s + (2.46 + 0.968i)7-s + (−2.09 − 1.21i)8-s + (−2.61 + 1.47i)9-s + (−0.793 + 0.458i)10-s + (0.216 + 0.0381i)11-s + (−2.16 − 1.52i)12-s + (−0.673 + 1.85i)13-s + (0.0464 − 1.81i)14-s + (−2.10 + 0.967i)15-s + (0.242 − 1.37i)16-s + (0.469 + 0.813i)17-s + ⋯ |
| L(s) = 1 | + (−0.166 − 0.456i)2-s + (−0.254 − 0.967i)3-s + (0.585 − 0.491i)4-s + (−0.103 − 0.588i)5-s + (−0.398 + 0.276i)6-s + (0.930 + 0.365i)7-s + (−0.741 − 0.428i)8-s + (−0.870 + 0.492i)9-s + (−0.251 + 0.144i)10-s + (0.0652 + 0.0115i)11-s + (−0.624 − 0.441i)12-s + (−0.186 + 0.513i)13-s + (0.0124 − 0.485i)14-s + (−0.542 + 0.249i)15-s + (0.0605 − 0.343i)16-s + (0.113 + 0.197i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.508 + 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.508 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.566018 - 0.991852i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.566018 - 0.991852i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.440 + 1.67i)T \) |
| 7 | \( 1 + (-2.46 - 0.968i)T \) |
| good | 2 | \( 1 + (0.234 + 0.645i)T + (-1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (0.231 + 1.31i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (-0.216 - 0.0381i)T + (10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (0.673 - 1.85i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.469 - 0.813i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.04 + 1.75i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.17 - 1.40i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (1.33 + 3.66i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-5.95 - 7.09i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (1.56 + 2.71i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-10.8 - 3.95i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.261 - 1.48i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (5.18 + 4.35i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 3.77iT - 53T^{2} \) |
| 59 | \( 1 + (-1.72 - 9.76i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-6.04 + 7.20i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (6.36 + 2.31i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-11.8 + 6.85i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (7.68 + 4.43i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7.91 - 2.88i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (16.1 - 5.88i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-3.46 + 5.99i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.42 - 0.780i)T + (91.1 + 33.1i)T^{2} \) |
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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
−12.08621961461455424402657537978, −11.42906700700425527773709128899, −10.60069545578497586973819692965, −9.140660743667685410311571674437, −8.218221483803030386379527311894, −7.01761287372707644599927381027, −5.97231963351221172395659957975, −4.80451072363744666353092171912, −2.48866540233521666925420881029, −1.26383387478182645234470946005,
2.79783590197167740484155610867, 4.16457832835839322369303861490, 5.53415164225617765189730333070, 6.72810517561718136909420032817, 7.83801319520904725453115950654, 8.713828895300804108185116819866, 10.11637886744237997090566730516, 11.01099609556546379173932317126, 11.54912313724787779355599715793, 12.72474704643207068616735883276
