L(s) = 1 | + (0.309 − 0.951i)2-s + (0.309 + 0.951i)3-s + (−0.809 − 0.587i)4-s − 5-s + 0.999·6-s + (−2.70 − 1.96i)7-s + (−0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + (−0.309 + 0.951i)10-s + (2.20 + 1.60i)11-s + (0.309 − 0.951i)12-s + (−0.0554 − 0.170i)13-s + (−2.70 + 1.96i)14-s + (−0.309 − 0.951i)15-s + (0.309 + 0.951i)16-s + (−0.444 + 0.322i)17-s + ⋯ |
L(s) = 1 | + (0.218 − 0.672i)2-s + (0.178 + 0.549i)3-s + (−0.404 − 0.293i)4-s − 0.447·5-s + 0.408·6-s + (−1.02 − 0.743i)7-s + (−0.286 + 0.207i)8-s + (−0.269 + 0.195i)9-s + (−0.0977 + 0.300i)10-s + (0.665 + 0.483i)11-s + (0.0892 − 0.274i)12-s + (−0.0153 − 0.0473i)13-s + (−0.723 + 0.525i)14-s + (−0.0797 − 0.245i)15-s + (0.0772 + 0.237i)16-s + (−0.107 + 0.0783i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.226 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.362328 + 0.456283i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.362328 + 0.456283i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.309 + 0.951i)T \) |
| 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + (0.685 - 5.52i)T \) |
good | 7 | \( 1 + (2.70 + 1.96i)T + (2.16 + 6.65i)T^{2} \) |
| 11 | \( 1 + (-2.20 - 1.60i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.0554 + 0.170i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (0.444 - 0.322i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (1.78 - 5.50i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (5.77 - 4.19i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.614 - 1.89i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + 4.34T + 37T^{2} \) |
| 41 | \( 1 + (1.99 - 6.14i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (0.575 - 1.77i)T + (-34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (1.46 + 4.51i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-3.17 + 2.30i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (0.199 + 0.613i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + 14.8T + 61T^{2} \) |
| 67 | \( 1 + 8.75T + 67T^{2} \) |
| 71 | \( 1 + (-8.57 + 6.23i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-7.43 - 5.40i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (5.66 - 4.11i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-1.98 + 6.12i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-0.842 - 0.612i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-2.54 - 1.85i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.16492078601683683843321516454, −9.803993136642168001125758957543, −8.864053350633308391565663641261, −7.896299849603792046955337310181, −6.85603705069825725866663355780, −5.93066853642345365338863614399, −4.68042790683302880026003520993, −3.77005614328862583652179704174, −3.34049811083734655774143634016, −1.68379059480303897281956784847,
0.24237805806458248703482017090, 2.39291729415706435644086157856, 3.46398622548532975211171586331, 4.48237947394565940701759098788, 5.82800913255062977389242164098, 6.39283086264760599435936164224, 7.13552796170697067820317908579, 8.127537486676184993664617056523, 8.906422323084128174556112843397, 9.427328122146751405170618362543