L(s) = 1 | + (1.28 + 2.21i)2-s + (−0.5 + 0.866i)3-s + (−2.28 + 3.95i)4-s + (1.28 + 2.21i)5-s − 2.56·6-s + (2.5 + 0.866i)7-s − 6.56·8-s + (−0.499 − 0.866i)9-s + (−3.28 + 5.68i)10-s + (1 − 1.73i)11-s + (−2.28 − 3.95i)12-s + 2.12·13-s + (1.28 + 6.65i)14-s − 2.56·15-s + (−3.84 − 6.65i)16-s + (1.71 − 2.97i)17-s + ⋯ |
L(s) = 1 | + (0.905 + 1.56i)2-s + (−0.288 + 0.499i)3-s + (−1.14 + 1.97i)4-s + (0.572 + 0.992i)5-s − 1.04·6-s + (0.944 + 0.327i)7-s − 2.31·8-s + (−0.166 − 0.288i)9-s + (−1.03 + 1.79i)10-s + (0.301 − 0.522i)11-s + (−0.658 − 1.14i)12-s + 0.588·13-s + (0.342 + 1.77i)14-s − 0.661·15-s + (−0.960 − 1.66i)16-s + (0.416 − 0.722i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.144812 - 2.28194i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.144812 - 2.28194i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (-1.28 - 2.21i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.28 - 2.21i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2.12T + 13T^{2} \) |
| 17 | \( 1 + (-1.71 + 2.97i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.78 + 4.81i)T + (-9.5 + 16.4i)T^{2} \) |
| 29 | \( 1 + 0.876T + 29T^{2} \) |
| 31 | \( 1 + (-3.34 + 5.78i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.78 + 6.54i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6.24T + 41T^{2} \) |
| 43 | \( 1 + 12.6T + 43T^{2} \) |
| 47 | \( 1 + (-2.84 - 4.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.40 - 11.0i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.438 - 0.759i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.93 + 5.08i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6.80T + 71T^{2} \) |
| 73 | \( 1 + (7.06 - 12.2i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.78 - 8.28i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 + (-5 - 8.66i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 9.36T + 97T^{2} \) |
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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
−11.40761081887226630971422848659, −10.75637206633647383231170114485, −9.373203386470896031017301014028, −8.517148262296334485792977703246, −7.54572658791151997267091225868, −6.54605009165360557606150031686, −5.93529181365315807535493468796, −5.04431152127747262566345262315, −4.09460737494508445701393598870, −2.80406520882602336697866095711,
1.36782226818114186719182331256, 1.78568754583796365940005761725, 3.60664737231626261627441002031, 4.67512297750776724756958899021, 5.34091302821197766296043342192, 6.36541954870547248708575720442, 8.059227927735918481623161656023, 8.928654162968726193263440888094, 10.13592590602564997231530472004, 10.60240803574409269414830699874