| L(s) = 1 | + (1.24 − 0.674i)2-s + (−1.63 − 0.585i)3-s + (1.09 − 1.67i)4-s + (−4.12 + 1.25i)5-s + (−2.42 + 0.371i)6-s + (−0.422 + 2.12i)7-s + (0.224 − 2.81i)8-s + (2.31 + 1.90i)9-s + (−4.28 + 4.33i)10-s + (−0.302 + 3.06i)11-s + (−2.75 + 2.09i)12-s + (0.171 + 0.0519i)13-s + (0.906 + 2.92i)14-s + (7.45 + 0.376i)15-s + (−1.62 − 3.65i)16-s + (−4.66 + 1.93i)17-s + ⋯ |
| L(s) = 1 | + (0.878 − 0.476i)2-s + (−0.941 − 0.338i)3-s + (0.545 − 0.838i)4-s + (−1.84 + 0.559i)5-s + (−0.988 + 0.151i)6-s + (−0.159 + 0.802i)7-s + (0.0795 − 0.996i)8-s + (0.771 + 0.636i)9-s + (−1.35 + 1.37i)10-s + (−0.0911 + 0.925i)11-s + (−0.796 + 0.604i)12-s + (0.0475 + 0.0144i)13-s + (0.242 + 0.781i)14-s + (1.92 + 0.0971i)15-s + (−0.405 − 0.914i)16-s + (−1.13 + 0.468i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.236 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.236 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.272006 + 0.345999i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.272006 + 0.345999i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.24 + 0.674i)T \) |
| 3 | \( 1 + (1.63 + 0.585i)T \) |
| good | 5 | \( 1 + (4.12 - 1.25i)T + (4.15 - 2.77i)T^{2} \) |
| 7 | \( 1 + (0.422 - 2.12i)T + (-6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (0.302 - 3.06i)T + (-10.7 - 2.14i)T^{2} \) |
| 13 | \( 1 + (-0.171 - 0.0519i)T + (10.8 + 7.22i)T^{2} \) |
| 17 | \( 1 + (4.66 - 1.93i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (3.10 - 1.66i)T + (10.5 - 15.7i)T^{2} \) |
| 23 | \( 1 + (1.45 - 2.18i)T + (-8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (0.0508 + 0.515i)T + (-28.4 + 5.65i)T^{2} \) |
| 31 | \( 1 + (6.38 - 6.38i)T - 31iT^{2} \) |
| 37 | \( 1 + (6.84 + 3.65i)T + (20.5 + 30.7i)T^{2} \) |
| 41 | \( 1 + (-6.61 + 9.89i)T + (-15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (1.77 + 2.15i)T + (-8.38 + 42.1i)T^{2} \) |
| 47 | \( 1 + (-1.93 + 0.801i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (0.566 - 5.74i)T + (-51.9 - 10.3i)T^{2} \) |
| 59 | \( 1 + (6.75 - 2.04i)T + (49.0 - 32.7i)T^{2} \) |
| 61 | \( 1 + (-0.642 + 0.782i)T + (-11.9 - 59.8i)T^{2} \) |
| 67 | \( 1 + (-4.09 - 3.35i)T + (13.0 + 65.7i)T^{2} \) |
| 71 | \( 1 + (2.20 - 11.0i)T + (-65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (2.42 - 0.482i)T + (67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (2.28 - 5.51i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-1.73 + 0.925i)T + (46.1 - 69.0i)T^{2} \) |
| 89 | \( 1 + (-8.86 + 5.92i)T + (34.0 - 82.2i)T^{2} \) |
| 97 | \( 1 + (5.12 + 5.12i)T + 97iT^{2} \) |
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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
−11.76672598339625308330492571401, −10.92051573235263452176138800518, −10.45727344327864057626183939824, −8.815648009415211807477214034935, −7.39519754022875943781218852913, −6.86973740194741673848876886279, −5.72069737826715539205788621564, −4.54041751448862224012333523324, −3.76553671890878021206211361935, −2.16335743093618235956273740057,
0.23747264258574221669920305970, 3.46329835531362477107197805783, 4.26924794328686957394504231220, 4.88339470115616846493102638153, 6.28179923423846940911886838637, 7.14450854086597133391809065388, 8.000833918719179777351090600550, 8.986155652523067412764447632575, 10.83618563248064677570259497145, 11.18709294407101576177953214959
