L(s) = 1 | + (0.555 − 0.831i)2-s + (−0.195 − 0.980i)3-s + (−0.382 − 0.923i)4-s + (0.831 − 0.555i)5-s + (−0.923 − 0.382i)6-s + (−0.980 − 0.195i)8-s + (−0.923 + 0.382i)9-s − i·10-s + (−0.831 + 0.555i)12-s + (−0.707 − 0.707i)15-s + (−0.707 + 0.707i)16-s + (1.17 − 1.17i)17-s + (−0.195 + 0.980i)18-s + (−1.08 + 1.63i)19-s + (−0.831 − 0.555i)20-s + ⋯ |
L(s) = 1 | + (0.555 − 0.831i)2-s + (−0.195 − 0.980i)3-s + (−0.382 − 0.923i)4-s + (0.831 − 0.555i)5-s + (−0.923 − 0.382i)6-s + (−0.980 − 0.195i)8-s + (−0.923 + 0.382i)9-s − i·10-s + (−0.831 + 0.555i)12-s + (−0.707 − 0.707i)15-s + (−0.707 + 0.707i)16-s + (1.17 − 1.17i)17-s + (−0.195 + 0.980i)18-s + (−1.08 + 1.63i)19-s + (−0.831 − 0.555i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.292721176\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.292721176\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.555 + 0.831i)T \) |
| 3 | \( 1 + (0.195 + 0.980i)T \) |
| 5 | \( 1 + (-0.831 + 0.555i)T \) |
good | 7 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 11 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 13 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 17 | \( 1 + (-1.17 + 1.17i)T - iT^{2} \) |
| 19 | \( 1 + (1.08 - 1.63i)T + (-0.382 - 0.923i)T^{2} \) |
| 23 | \( 1 + (-0.425 - 1.02i)T + (-0.707 + 0.707i)T^{2} \) |
| 29 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 31 | \( 1 + 0.765iT - T^{2} \) |
| 37 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 41 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 43 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 47 | \( 1 + (-0.275 + 0.275i)T - iT^{2} \) |
| 53 | \( 1 + (1.81 + 0.360i)T + (0.923 + 0.382i)T^{2} \) |
| 59 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 61 | \( 1 + (-0.382 - 1.92i)T + (-0.923 + 0.382i)T^{2} \) |
| 67 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 71 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 73 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 79 | \( 1 + (1 + i)T + iT^{2} \) |
| 83 | \( 1 + (-1.02 + 1.53i)T + (-0.382 - 0.923i)T^{2} \) |
| 89 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.965491606336471624617478963180, −9.243770075815466407562939611776, −8.287896620920392887047723378145, −7.27308048414387873998525248243, −5.99067964880803967488275780316, −5.70466426910015421227365222059, −4.66090359967582869081791427803, −3.27092926322866291368622590960, −2.13318016414910592432695541879, −1.19654697643366367236649046076,
2.59884759174056316058842349841, 3.54344266915375579986994273987, 4.62119316172468164648239546315, 5.37835988812435257252656189439, 6.27544062892016241740102925169, 6.81666497163371116754287495398, 8.154737443266736493968524643886, 8.896999266786003001183238410260, 9.670778984846052233575019938063, 10.60844894063203309346139686748