L(s) = 1 | + (0.965 − 0.258i)2-s + (0.563 − 1.63i)3-s + (0.866 − 0.499i)4-s + (−1.58 − 1.58i)5-s + (0.120 − 1.72i)6-s + (−0.258 + 0.965i)7-s + (0.707 − 0.707i)8-s + (−2.36 − 1.84i)9-s + (−1.93 − 1.11i)10-s + (−1.28 − 4.80i)11-s + (−0.331 − 1.70i)12-s + (0.738 + 3.52i)13-s + i·14-s + (−3.48 + 1.70i)15-s + (0.500 − 0.866i)16-s + (−1.25 − 2.17i)17-s + ⋯ |
L(s) = 1 | + (0.683 − 0.183i)2-s + (0.325 − 0.945i)3-s + (0.433 − 0.249i)4-s + (−0.707 − 0.707i)5-s + (0.0490 − 0.705i)6-s + (−0.0978 + 0.365i)7-s + (0.249 − 0.249i)8-s + (−0.788 − 0.614i)9-s + (−0.613 − 0.353i)10-s + (−0.388 − 1.44i)11-s + (−0.0956 − 0.490i)12-s + (0.204 + 0.978i)13-s + 0.267i·14-s + (−0.899 + 0.439i)15-s + (0.125 − 0.216i)16-s + (−0.303 − 0.526i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.634 + 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.810404 - 1.71303i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.810404 - 1.71303i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 + 0.258i)T \) |
| 3 | \( 1 + (-0.563 + 1.63i)T \) |
| 7 | \( 1 + (0.258 - 0.965i)T \) |
| 13 | \( 1 + (-0.738 - 3.52i)T \) |
good | 5 | \( 1 + (1.58 + 1.58i)T + 5iT^{2} \) |
| 11 | \( 1 + (1.28 + 4.80i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (1.25 + 2.17i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.54 - 0.412i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-0.566 + 0.981i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.50 - 3.18i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.36 + 1.36i)T - 31iT^{2} \) |
| 37 | \( 1 + (-1.84 + 0.494i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-3.36 + 0.902i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-6.80 + 3.92i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (8.51 - 8.51i)T - 47iT^{2} \) |
| 53 | \( 1 + 1.61iT - 53T^{2} \) |
| 59 | \( 1 + (-5.35 - 1.43i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-1.56 - 2.71i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.18 + 11.8i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-0.709 + 2.64i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-8.64 - 8.64i)T + 73iT^{2} \) |
| 79 | \( 1 - 17.5T + 79T^{2} \) |
| 83 | \( 1 + (-4.87 - 4.87i)T + 83iT^{2} \) |
| 89 | \( 1 + (1.28 + 4.78i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (15.8 + 4.25i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.96043868057972552026379110380, −9.340486389738065507720942862614, −8.564474966798669794590994197269, −7.86617870445332928149046026962, −6.71924900135781006296454976437, −5.91464555431988483489167743091, −4.78898404351233464659239423366, −3.56247604774189599369799508445, −2.50859717253581797554159921065, −0.866786708019047538895542473170,
2.55705849747117594586701563563, 3.53285533071032487527477372227, 4.39423427101245318152554380945, 5.28389857679752591942927688333, 6.57749081231754342296733870266, 7.58600501220476688520152298107, 8.187637523975347921683372605970, 9.600350453790178424596213891488, 10.37276067343232470827555001075, 10.99034296772687531048664300371