| L(s) = 1 | + (1.20 − 1.68i)2-s + (0.235 − 0.971i)3-s + (−0.747 − 2.15i)4-s + (−0.0741 + 1.55i)5-s + (−1.35 − 1.56i)6-s + (2.38 − 1.15i)7-s + (−0.566 − 0.166i)8-s + (−0.888 − 0.458i)9-s + (2.53 + 1.99i)10-s + (−1.77 − 2.49i)11-s + (−2.27 + 0.217i)12-s + (0.522 − 3.63i)13-s + (0.913 − 5.40i)14-s + (1.49 + 0.438i)15-s + (2.63 − 2.06i)16-s + (−2.78 + 0.535i)17-s + ⋯ |
| L(s) = 1 | + (0.849 − 1.19i)2-s + (0.136 − 0.561i)3-s + (−0.373 − 1.07i)4-s + (−0.0331 + 0.695i)5-s + (−0.553 − 0.638i)6-s + (0.899 − 0.436i)7-s + (−0.200 − 0.0588i)8-s + (−0.296 − 0.152i)9-s + (0.801 + 0.630i)10-s + (−0.536 − 0.752i)11-s + (−0.656 + 0.0627i)12-s + (0.144 − 1.00i)13-s + (0.244 − 1.44i)14-s + (0.385 + 0.113i)15-s + (0.657 − 0.517i)16-s + (−0.674 + 0.129i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.559 + 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.12600 - 2.11990i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12600 - 2.11990i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.235 + 0.971i)T \) |
| 7 | \( 1 + (-2.38 + 1.15i)T \) |
| 23 | \( 1 + (-2.76 - 3.92i)T \) |
| good | 2 | \( 1 + (-1.20 + 1.68i)T + (-0.654 - 1.89i)T^{2} \) |
| 5 | \( 1 + (0.0741 - 1.55i)T + (-4.97 - 0.475i)T^{2} \) |
| 11 | \( 1 + (1.77 + 2.49i)T + (-3.59 + 10.3i)T^{2} \) |
| 13 | \( 1 + (-0.522 + 3.63i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (2.78 - 0.535i)T + (15.7 - 6.31i)T^{2} \) |
| 19 | \( 1 + (2.94 + 0.567i)T + (17.6 + 7.06i)T^{2} \) |
| 29 | \( 1 + (-4.38 - 5.06i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (6.78 - 6.47i)T + (1.47 - 30.9i)T^{2} \) |
| 37 | \( 1 + (1.88 + 0.969i)T + (21.4 + 30.1i)T^{2} \) |
| 41 | \( 1 + (-4.01 - 2.57i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-2.31 + 0.678i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (-5.81 - 10.0i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (10.4 + 4.18i)T + (38.3 + 36.5i)T^{2} \) |
| 59 | \( 1 + (-4.53 - 3.56i)T + (13.9 + 57.3i)T^{2} \) |
| 61 | \( 1 + (-1.21 - 4.99i)T + (-54.2 + 27.9i)T^{2} \) |
| 67 | \( 1 + (7.43 + 0.709i)T + (65.7 + 12.6i)T^{2} \) |
| 71 | \( 1 + (-2.84 - 6.22i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (3.82 + 11.0i)T + (-57.3 + 45.1i)T^{2} \) |
| 79 | \( 1 + (-8.12 + 3.25i)T + (57.1 - 54.5i)T^{2} \) |
| 83 | \( 1 + (14.0 - 9.04i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (0.639 + 0.609i)T + (4.23 + 88.8i)T^{2} \) |
| 97 | \( 1 + (-6.82 - 4.38i)T + (40.2 + 88.2i)T^{2} \) |
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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
−10.77503114522432606793504231053, −10.56726510230656580114160567395, −8.884192566641811174895837795801, −7.896384241927674915163402807066, −7.03808957385697693317765679395, −5.68506771821058101358071642310, −4.74309391597648383169294034001, −3.43159741876277927751878798464, −2.66806567714871992298062940578, −1.29879247792788220423806773905,
2.16706558366126319934573545479, 4.23542904651068990748496108276, 4.61877172651985831329714462569, 5.46772053779918998550872170206, 6.54397753267198921412023928463, 7.57616598918932811479782579739, 8.492518859266530436581457090118, 9.134876558062477045502567625172, 10.43224817053647946856298240543, 11.36122670351549359885394352982
