| L(s) = 1 | + (−1.21 + 0.720i)2-s + (1.59 + 1.92i)3-s + (0.961 − 1.75i)4-s + (−2.00 + 0.991i)5-s + (−3.32 − 1.19i)6-s + (−0.979 + 0.318i)7-s + (0.0943 + 2.82i)8-s + (−0.605 + 3.17i)9-s + (1.72 − 2.65i)10-s + (0.568 − 4.49i)11-s + (4.90 − 0.941i)12-s + (0.649 − 3.40i)13-s + (0.962 − 1.09i)14-s + (−5.09 − 2.27i)15-s + (−2.15 − 3.37i)16-s + (−1.06 − 4.13i)17-s + ⋯ |
| L(s) = 1 | + (−0.860 + 0.509i)2-s + (0.918 + 1.11i)3-s + (0.480 − 0.876i)4-s + (−0.896 + 0.443i)5-s + (−1.35 − 0.487i)6-s + (−0.370 + 0.120i)7-s + (0.0333 + 0.999i)8-s + (−0.201 + 1.05i)9-s + (0.545 − 0.838i)10-s + (0.171 − 1.35i)11-s + (1.41 − 0.271i)12-s + (0.180 − 0.944i)13-s + (0.257 − 0.292i)14-s + (−1.31 − 0.587i)15-s + (−0.538 − 0.842i)16-s + (−0.257 − 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.447925 - 0.226985i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.447925 - 0.226985i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.21 - 0.720i)T \) |
| 5 | \( 1 + (2.00 - 0.991i)T \) |
| good | 3 | \( 1 + (-1.59 - 1.92i)T + (-0.562 + 2.94i)T^{2} \) |
| 7 | \( 1 + (0.979 - 0.318i)T + (5.66 - 4.11i)T^{2} \) |
| 11 | \( 1 + (-0.568 + 4.49i)T + (-10.6 - 2.73i)T^{2} \) |
| 13 | \( 1 + (-0.649 + 3.40i)T + (-12.0 - 4.78i)T^{2} \) |
| 17 | \( 1 + (1.06 + 4.13i)T + (-14.8 + 8.18i)T^{2} \) |
| 19 | \( 1 + (6.24 + 5.16i)T + (3.56 + 18.6i)T^{2} \) |
| 23 | \( 1 + (4.52 - 4.81i)T + (-1.44 - 22.9i)T^{2} \) |
| 29 | \( 1 + (7.60 - 0.478i)T + (28.7 - 3.63i)T^{2} \) |
| 31 | \( 1 + (-5.09 + 1.30i)T + (27.1 - 14.9i)T^{2} \) |
| 37 | \( 1 + (3.98 + 2.19i)T + (19.8 + 31.2i)T^{2} \) |
| 41 | \( 1 + (0.0408 - 0.0383i)T + (2.57 - 40.9i)T^{2} \) |
| 43 | \( 1 + (-0.790 + 0.574i)T + (13.2 - 40.8i)T^{2} \) |
| 47 | \( 1 + (-2.29 - 5.80i)T + (-34.2 + 32.1i)T^{2} \) |
| 53 | \( 1 + (-0.100 + 0.158i)T + (-22.5 - 47.9i)T^{2} \) |
| 59 | \( 1 + (1.33 - 0.630i)T + (37.6 - 45.4i)T^{2} \) |
| 61 | \( 1 + (0.797 - 0.849i)T + (-3.83 - 60.8i)T^{2} \) |
| 67 | \( 1 + (0.0173 - 0.276i)T + (-66.4 - 8.39i)T^{2} \) |
| 71 | \( 1 + (3.75 - 1.48i)T + (51.7 - 48.6i)T^{2} \) |
| 73 | \( 1 + (-9.47 - 4.45i)T + (46.5 + 56.2i)T^{2} \) |
| 79 | \( 1 + (7.03 + 8.50i)T + (-14.8 + 77.6i)T^{2} \) |
| 83 | \( 1 + (-11.1 + 13.4i)T + (-15.5 - 81.5i)T^{2} \) |
| 89 | \( 1 + (-0.167 + 0.355i)T + (-56.7 - 68.5i)T^{2} \) |
| 97 | \( 1 + (11.3 - 0.714i)T + (96.2 - 12.1i)T^{2} \) |
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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
−9.586474240360991399045683243971, −8.998406298924638143299993201689, −8.323602326131389180942788272762, −7.67748281879203406674465137560, −6.61735009570289072161021212730, −5.65376791287349257751739922036, −4.43513461698326836208433175433, −3.38088438080650627887638075523, −2.63771499230840428712754491400, −0.26047256422742170914004681192,
1.62590605170013386049720832338, 2.19147333565540342933478816957, 3.72247197940484359468566085727, 4.27991419802349419835878466274, 6.49095350094258349030599850023, 6.93563257644862335072028935282, 7.957623809175329144034343773448, 8.295486047908377652657702659072, 9.045937434754784011557686503802, 9.970173270945750506949721497056
