| L(s) = 1 | + (−1.04 − 0.954i)2-s + (1.68 + 0.409i)3-s + (0.179 + 1.99i)4-s + (0.0417 + 0.0624i)5-s + (−1.36 − 2.03i)6-s + (−0.100 + 0.242i)7-s + (1.71 − 2.25i)8-s + (2.66 + 1.37i)9-s + (0.0160 − 0.104i)10-s + (2.15 + 0.428i)11-s + (−0.514 + 3.42i)12-s + (0.751 + 0.502i)13-s + (0.335 − 0.157i)14-s + (0.0445 + 0.122i)15-s + (−3.93 + 0.714i)16-s + (1.66 + 1.66i)17-s + ⋯ |
| L(s) = 1 | + (−0.738 − 0.674i)2-s + (0.971 + 0.236i)3-s + (0.0896 + 0.995i)4-s + (0.0186 + 0.0279i)5-s + (−0.557 − 0.830i)6-s + (−0.0379 + 0.0915i)7-s + (0.605 − 0.795i)8-s + (0.887 + 0.459i)9-s + (0.00506 − 0.0331i)10-s + (0.650 + 0.129i)11-s + (−0.148 + 0.988i)12-s + (0.208 + 0.139i)13-s + (0.0897 − 0.0419i)14-s + (0.0115 + 0.0315i)15-s + (−0.983 + 0.178i)16-s + (0.404 + 0.404i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.15226 - 0.153112i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15226 - 0.153112i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.04 + 0.954i)T \) |
| 3 | \( 1 + (-1.68 - 0.409i)T \) |
| good | 5 | \( 1 + (-0.0417 - 0.0624i)T + (-1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (0.100 - 0.242i)T + (-4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-2.15 - 0.428i)T + (10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (-0.751 - 0.502i)T + (4.97 + 12.0i)T^{2} \) |
| 17 | \( 1 + (-1.66 - 1.66i)T + 17iT^{2} \) |
| 19 | \( 1 + (1.06 + 0.711i)T + (7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (0.745 + 1.79i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (1.26 + 6.37i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 + (-2.54 - 3.81i)T + (-14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (4.77 - 1.97i)T + (28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (9.37 + 1.86i)T + (39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (8.04 - 8.04i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1.29 + 6.50i)T + (-48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (3.86 - 2.57i)T + (22.5 - 54.5i)T^{2} \) |
| 61 | \( 1 + (0.853 + 4.29i)T + (-56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 + (-1.79 + 0.357i)T + (61.8 - 25.6i)T^{2} \) |
| 71 | \( 1 + (5.21 + 2.16i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-10.0 + 4.15i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (5.93 - 5.93i)T - 79iT^{2} \) |
| 83 | \( 1 + (-6.79 + 10.1i)T + (-31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (13.5 + 5.62i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 - 13.0iT - 97T^{2} \) |
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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
−12.47176118773849618957239242770, −11.38119786157224084560885649432, −10.30086999956222926212618997087, −9.519667408241918879362689756163, −8.634707107461949315081008874861, −7.82006057617615135460680842334, −6.59335660891228562639466420748, −4.40483822240316065315564609760, −3.29958680240239825956658303356, −1.86443759989596285120920605918,
1.62416270592390354792297742996, 3.56848990707789563295173894504, 5.32570147398171134951117697235, 6.76575364936202117968089081709, 7.50790656075664310207289433918, 8.639805848477598489559862794567, 9.272957513801456853276113548274, 10.26185410562211587514620648874, 11.43249178052264947079566640150, 12.81559992549880492398851210117
