L(s) = 1 | + (−0.939 + 0.342i)2-s + (−0.326 + 1.85i)3-s + (0.766 − 0.642i)4-s + (1.53 + 1.28i)5-s + (−0.326 − 1.85i)6-s + (−2.53 − 4.38i)7-s + (−0.500 + 0.866i)8-s + (−0.5 − 0.181i)9-s + (−1.87 − 0.684i)10-s + (0.705 − 1.22i)11-s + (0.939 + 1.62i)12-s + (−0.226 − 1.28i)13-s + (3.87 + 3.25i)14-s + (−2.87 + 2.41i)15-s + (0.173 − 0.984i)16-s + (−2.24 + 0.817i)17-s + ⋯ |
L(s) = 1 | + (−0.664 + 0.241i)2-s + (−0.188 + 1.06i)3-s + (0.383 − 0.321i)4-s + (0.685 + 0.574i)5-s + (−0.133 − 0.755i)6-s + (−0.957 − 1.65i)7-s + (−0.176 + 0.306i)8-s + (−0.166 − 0.0606i)9-s + (−0.594 − 0.216i)10-s + (0.212 − 0.368i)11-s + (0.271 + 0.469i)12-s + (−0.0628 − 0.356i)13-s + (1.03 + 0.869i)14-s + (−0.743 + 0.623i)15-s + (0.0434 − 0.246i)16-s + (−0.544 + 0.198i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.518203 + 0.258727i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.518203 + 0.258727i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.939 - 0.342i)T \) |
| 19 | \( 1 + (-2.23 - 3.74i)T \) |
good | 3 | \( 1 + (0.326 - 1.85i)T + (-2.81 - 1.02i)T^{2} \) |
| 5 | \( 1 + (-1.53 - 1.28i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (2.53 + 4.38i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.705 + 1.22i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.226 + 1.28i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (2.24 - 0.817i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (2.34 - 1.96i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (7.94 + 2.89i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (0.184 + 0.320i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.82T + 37T^{2} \) |
| 41 | \( 1 + (0.266 - 1.50i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (0.581 + 0.487i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-9.59 - 3.49i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-1.28 + 1.07i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (0.673 - 0.245i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-7.47 + 6.27i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-1.31 - 0.480i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (4.87 + 4.09i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (0.791 - 4.49i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (0.389 - 2.20i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.99 - 3.45i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1.84 + 10.4i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-1.43 + 0.524i)T + (74.3 - 62.3i)T^{2} \) |
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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
−16.60316683238768635664897147692, −15.73535174470591341542131140447, −14.35810197406338571237907968148, −13.26279022936474394476582380031, −11.01318144367772548069113059508, −10.16482174993399855833094476948, −9.585259334932711390944439951638, −7.45439198546059197173016381339, −6.06254036616355721770104858340, −3.82543604641660863103117876472,
2.19412227243631258224168108529, 5.79828291177037356328854112871, 7.03458196380242232525338150663, 8.890381628757757005395721501908, 9.585247188339720934620358942217, 11.67144201386471282588377694036, 12.57271204960983390965263945601, 13.30779058188812224841608757252, 15.27564374915004154227256740969, 16.41995344504839540696850991715