L(s) = 1 | + (−0.879 − 1.10i)2-s + (1.68 − 0.405i)3-s + (−0.452 + 1.94i)4-s + (−0.324 − 3.29i)5-s + (−1.93 − 1.50i)6-s + (−3.28 − 2.19i)7-s + (2.55 − 1.21i)8-s + (2.67 − 1.36i)9-s + (−3.36 + 3.25i)10-s + (−0.281 − 0.525i)11-s + (0.0294 + 3.46i)12-s + (−0.187 + 1.90i)13-s + (0.459 + 5.57i)14-s + (−1.88 − 5.41i)15-s + (−3.59 − 1.76i)16-s + (−2.42 + 5.84i)17-s + ⋯ |
L(s) = 1 | + (−0.622 − 0.782i)2-s + (0.972 − 0.234i)3-s + (−0.226 + 0.974i)4-s + (−0.145 − 1.47i)5-s + (−0.788 − 0.615i)6-s + (−1.24 − 0.829i)7-s + (0.903 − 0.428i)8-s + (0.890 − 0.455i)9-s + (−1.06 + 1.02i)10-s + (−0.0847 − 0.158i)11-s + (0.00850 + 0.999i)12-s + (−0.0521 + 0.529i)13-s + (0.122 + 1.48i)14-s + (−0.486 − 1.39i)15-s + (−0.897 − 0.440i)16-s + (−0.587 + 1.41i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 + 0.362i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.931 + 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.188379 - 1.00387i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.188379 - 1.00387i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.879 + 1.10i)T \) |
| 3 | \( 1 + (-1.68 + 0.405i)T \) |
good | 5 | \( 1 + (0.324 + 3.29i)T + (-4.90 + 0.975i)T^{2} \) |
| 7 | \( 1 + (3.28 + 2.19i)T + (2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (0.281 + 0.525i)T + (-6.11 + 9.14i)T^{2} \) |
| 13 | \( 1 + (0.187 - 1.90i)T + (-12.7 - 2.53i)T^{2} \) |
| 17 | \( 1 + (2.42 - 5.84i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (1.15 - 0.948i)T + (3.70 - 18.6i)T^{2} \) |
| 23 | \( 1 + (-1.22 + 6.16i)T + (-21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (-3.15 + 5.90i)T + (-16.1 - 24.1i)T^{2} \) |
| 31 | \( 1 + (1.04 + 1.04i)T + 31iT^{2} \) |
| 37 | \( 1 + (-0.353 - 0.290i)T + (7.21 + 36.2i)T^{2} \) |
| 41 | \( 1 + (-2.35 + 11.8i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (-0.626 + 2.06i)T + (-35.7 - 23.8i)T^{2} \) |
| 47 | \( 1 + (-0.749 + 1.80i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-5.62 - 10.5i)T + (-29.4 + 44.0i)T^{2} \) |
| 59 | \( 1 + (0.520 + 5.28i)T + (-57.8 + 11.5i)T^{2} \) |
| 61 | \( 1 + (-3.82 - 12.6i)T + (-50.7 + 33.8i)T^{2} \) |
| 67 | \( 1 + (-4.28 + 1.29i)T + (55.7 - 37.2i)T^{2} \) |
| 71 | \( 1 + (5.35 + 3.57i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-5.77 - 8.64i)T + (-27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-11.1 + 4.62i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (2.60 - 2.13i)T + (16.1 - 81.4i)T^{2} \) |
| 89 | \( 1 + (-12.0 + 2.39i)T + (82.2 - 34.0i)T^{2} \) |
| 97 | \( 1 + (-5.23 + 5.23i)T - 97iT^{2} \) |
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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
−10.61419937198326611884223847051, −9.931885068688423413436014192309, −8.910511639525011726309971118853, −8.608567690761886059710218464036, −7.53674500844144473161080121018, −6.44374821449545990343457161082, −4.31270718728740627971655006707, −3.82860580011027040414839585378, −2.25502750468278142770564017501, −0.74510024471982432717442324776,
2.51786046722952698824020318492, 3.32489174871415271784892748569, 5.10302809326445865864923552289, 6.48900325621783484955527982711, 7.03952922441722638550095631979, 7.950809544300454325801768632325, 9.157174811645703204661689918521, 9.628854596587129685544851830192, 10.46236703526673117870754968996, 11.40778065126405638093209706651