L(s) = 1 | + (−0.951 + 0.309i)2-s + (0.951 + 0.309i)3-s + (0.809 − 0.587i)4-s − 2.23·5-s − 0.999·6-s + (−0.726 − i)7-s + (−0.587 + 0.809i)8-s + (0.809 + 0.587i)9-s + (2.12 − 0.690i)10-s + (1.92 − 1.40i)11-s + (0.951 − 0.309i)12-s + (−0.138 − 0.0450i)13-s + (1 + 0.726i)14-s + (−2.12 − 0.690i)15-s + (0.309 − 0.951i)16-s + (−1.08 + 1.5i)17-s + ⋯ |
L(s) = 1 | + (−0.672 + 0.218i)2-s + (0.549 + 0.178i)3-s + (0.404 − 0.293i)4-s − 0.999·5-s − 0.408·6-s + (−0.274 − 0.377i)7-s + (−0.207 + 0.286i)8-s + (0.269 + 0.195i)9-s + (0.672 − 0.218i)10-s + (0.581 − 0.422i)11-s + (0.274 − 0.0892i)12-s + (−0.0384 − 0.0125i)13-s + (0.267 + 0.194i)14-s + (−0.549 − 0.178i)15-s + (0.0772 − 0.237i)16-s + (−0.264 + 0.363i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0525 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0525 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.534161 - 0.506802i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.534161 - 0.506802i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 3 | \( 1 + (-0.951 - 0.309i)T \) |
| 5 | \( 1 + 2.23T \) |
| 31 | \( 1 + (2.19 + 5.11i)T \) |
good | 7 | \( 1 + (0.726 + i)T + (-2.16 + 6.65i)T^{2} \) |
| 11 | \( 1 + (-1.92 + 1.40i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (0.138 + 0.0450i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (1.08 - 1.5i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (1.38 + 4.25i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (3.44 - 4.73i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (2.11 + 6.51i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + 10.8iT - 37T^{2} \) |
| 41 | \( 1 + (2.38 + 7.33i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-3.80 + 1.23i)T + (34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (2.93 + 0.954i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (4.53 - 6.23i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.972 + 2.99i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 - 2.76T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + (-3.23 - 2.35i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.73 - 2.38i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-4.92 - 3.57i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (2.62 - 0.854i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-1.85 + 1.34i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.555 - 0.763i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−9.599164397863877359045389292264, −9.045969807496352198704981036171, −8.173027287246235022972699703070, −7.50745311157730221938456336001, −6.76498642185410889666463018560, −5.65256835733481980690506468997, −4.19643222037304653192584655764, −3.59890339842053591187351694823, −2.18193084488975344559714906387, −0.41329827043486249544429323964,
1.47320422078745079868641460257, 2.82959794689358377945372871596, 3.73508892047316508612923914146, 4.77147700683066340496865186143, 6.37165898323396694379267928418, 7.02496168125096623170745664535, 8.004089990531486568821638177870, 8.526015989366627662848348540692, 9.328360850936659872396809548585, 10.15554630313362662982839458632