| L(s) = 1 | + (1.41 + 0.0401i)2-s + (0.980 − 0.195i)3-s + (1.99 + 0.113i)4-s + (−1.10 − 1.66i)5-s + (1.39 − 0.236i)6-s + (−1.97 − 0.816i)7-s + (2.81 + 0.240i)8-s + (0.923 − 0.382i)9-s + (−1.50 − 2.39i)10-s + (−0.842 + 4.23i)11-s + (1.98 − 0.278i)12-s + (0.0329 − 0.0493i)13-s + (−2.75 − 1.23i)14-s + (−1.41 − 1.41i)15-s + (3.97 + 0.453i)16-s + (−5.58 + 5.58i)17-s + ⋯ |
| L(s) = 1 | + (0.999 + 0.0284i)2-s + (0.566 − 0.112i)3-s + (0.998 + 0.0568i)4-s + (−0.496 − 0.742i)5-s + (0.569 − 0.0964i)6-s + (−0.744 − 0.308i)7-s + (0.996 + 0.0851i)8-s + (0.307 − 0.127i)9-s + (−0.475 − 0.756i)10-s + (−0.254 + 1.27i)11-s + (0.571 − 0.0802i)12-s + (0.00914 − 0.0136i)13-s + (−0.735 − 0.329i)14-s + (−0.364 − 0.364i)15-s + (0.993 + 0.113i)16-s + (−1.35 + 1.35i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.12536 - 0.287280i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.12536 - 0.287280i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.41 - 0.0401i)T \) |
| 3 | \( 1 + (-0.980 + 0.195i)T \) |
| good | 5 | \( 1 + (1.10 + 1.66i)T + (-1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (1.97 + 0.816i)T + (4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (0.842 - 4.23i)T + (-10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (-0.0329 + 0.0493i)T + (-4.97 - 12.0i)T^{2} \) |
| 17 | \( 1 + (5.58 - 5.58i)T - 17iT^{2} \) |
| 19 | \( 1 + (-0.497 - 0.332i)T + (7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (-0.591 - 1.42i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (1.85 + 9.31i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 - 2.71iT - 31T^{2} \) |
| 37 | \( 1 + (-7.23 + 4.83i)T + (14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (3.74 + 9.02i)T + (-28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (-1.17 - 0.234i)T + (39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (6.16 - 6.16i)T - 47iT^{2} \) |
| 53 | \( 1 + (0.912 - 4.58i)T + (-48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (-1.50 - 2.25i)T + (-22.5 + 54.5i)T^{2} \) |
| 61 | \( 1 + (2.18 - 0.434i)T + (56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + (-14.1 + 2.80i)T + (61.8 - 25.6i)T^{2} \) |
| 71 | \( 1 + (0.535 + 0.221i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (2.44 - 1.01i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (6.21 + 6.21i)T + 79iT^{2} \) |
| 83 | \( 1 + (-4.16 - 2.78i)T + (31.7 + 76.6i)T^{2} \) |
| 89 | \( 1 + (0.479 - 1.15i)T + (-62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 - 0.928iT - 97T^{2} \) |
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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
−12.93559833268183857838145925904, −11.90527563668888275874567204212, −10.64384868045852832523154518316, −9.571970615210285962499715433024, −8.246821397080924675011273773442, −7.26976649902487843236694842379, −6.20480332260809578222820683549, −4.60145805069165273957185456833, −3.85585455308775781956475847127, −2.17163487405731760272271478642,
2.76715208608988357030418385402, 3.41805668256924962428202769128, 4.93280619326880792925984544036, 6.37916295131960944380204133996, 7.15764105875533061439496401147, 8.448897650411289370448216718951, 9.708116989416660986774247902570, 11.06105647549162812382992098523, 11.45060195364385447742672916683, 12.90987036411711229693834737609
