| L(s) = 1 | + (−1.40 − 0.159i)2-s + (2.35 − 0.713i)3-s + (1.94 + 0.447i)4-s + (−0.995 + 0.0980i)5-s + (−3.41 + 0.627i)6-s + (1.13 − 1.70i)7-s + (−2.66 − 0.939i)8-s + (2.52 − 1.68i)9-s + (1.41 + 0.0207i)10-s + (−2.93 − 5.49i)11-s + (4.90 − 0.338i)12-s + (−0.102 + 1.04i)13-s + (−1.86 + 2.20i)14-s + (−2.26 + 0.940i)15-s + (3.59 + 1.74i)16-s + (−4.25 − 1.76i)17-s + ⋯ |
| L(s) = 1 | + (−0.993 − 0.112i)2-s + (1.35 − 0.411i)3-s + (0.974 + 0.223i)4-s + (−0.445 + 0.0438i)5-s + (−1.39 + 0.256i)6-s + (0.429 − 0.642i)7-s + (−0.943 − 0.332i)8-s + (0.841 − 0.562i)9-s + (0.447 + 0.00655i)10-s + (−0.886 − 1.65i)11-s + (1.41 − 0.0976i)12-s + (−0.0284 + 0.288i)13-s + (−0.499 + 0.590i)14-s + (−0.586 + 0.242i)15-s + (0.899 + 0.436i)16-s + (−1.03 − 0.427i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.816901 - 0.951234i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.816901 - 0.951234i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.40 + 0.159i)T \) |
| 5 | \( 1 + (0.995 - 0.0980i)T \) |
| good | 3 | \( 1 + (-2.35 + 0.713i)T + (2.49 - 1.66i)T^{2} \) |
| 7 | \( 1 + (-1.13 + 1.70i)T + (-2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (2.93 + 5.49i)T + (-6.11 + 9.14i)T^{2} \) |
| 13 | \( 1 + (0.102 - 1.04i)T + (-12.7 - 2.53i)T^{2} \) |
| 17 | \( 1 + (4.25 + 1.76i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (1.36 + 1.65i)T + (-3.70 + 18.6i)T^{2} \) |
| 23 | \( 1 + (-1.41 + 7.11i)T + (-21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (-6.38 - 3.41i)T + (16.1 + 24.1i)T^{2} \) |
| 31 | \( 1 + (-1.95 + 1.95i)T - 31iT^{2} \) |
| 37 | \( 1 + (-5.50 - 4.51i)T + (7.21 + 36.2i)T^{2} \) |
| 41 | \( 1 + (-5.52 - 1.09i)T + (37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (3.63 + 1.10i)T + (35.7 + 23.8i)T^{2} \) |
| 47 | \( 1 + (0.728 - 1.75i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-3.74 + 2.00i)T + (29.4 - 44.0i)T^{2} \) |
| 59 | \( 1 + (-0.348 - 3.53i)T + (-57.8 + 11.5i)T^{2} \) |
| 61 | \( 1 + (-3.20 - 10.5i)T + (-50.7 + 33.8i)T^{2} \) |
| 67 | \( 1 + (0.970 + 3.19i)T + (-55.7 + 37.2i)T^{2} \) |
| 71 | \( 1 + (-5.35 - 3.57i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (6.07 + 9.09i)T + (-27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (4.00 + 9.66i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-10.1 + 8.34i)T + (16.1 - 81.4i)T^{2} \) |
| 89 | \( 1 + (0.316 + 1.59i)T + (-82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (-4.24 + 4.24i)T - 97iT^{2} \) |
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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
−10.39650049126257404796735974025, −9.045581643792670358008615444798, −8.539383869045877185650893881300, −8.017204618266468939812943727028, −7.19251375613567969910518081638, −6.29035932955853256501507896792, −4.48333900884667825314879864814, −3.09836438199646539447429276436, −2.47802651453133044511115479711, −0.77976006341920382739933364647,
1.99713732447952513952021411625, 2.69898261084173785583410837885, 4.10394862032784508036639467784, 5.29934167089926544347122555075, 6.77378622680972378762042849984, 7.82985506855893598086746501639, 8.132398428644532775698638378332, 9.063057138152726954600544595620, 9.717624980914950903791247609056, 10.45082293828593899295070088292
