L(s) = 1 | + (−1.38 − 0.300i)2-s + (1.24 + 1.20i)3-s + (1.81 + 0.830i)4-s + 2.72i·5-s + (−1.35 − 2.03i)6-s + (−2.26 − 1.69i)8-s + (0.0992 + 2.99i)9-s + (0.819 − 3.76i)10-s − 2.04·11-s + (1.26 + 3.22i)12-s + 4.44·13-s + (−3.28 + 3.39i)15-s + (2.62 + 3.02i)16-s + 0.660i·17-s + (0.763 − 4.17i)18-s + 2.93i·19-s + ⋯ |
L(s) = 1 | + (−0.977 − 0.212i)2-s + (0.718 + 0.695i)3-s + (0.909 + 0.415i)4-s + 1.21i·5-s + (−0.554 − 0.832i)6-s + (−0.800 − 0.598i)8-s + (0.0330 + 0.999i)9-s + (0.259 − 1.19i)10-s − 0.615·11-s + (0.365 + 0.930i)12-s + 1.23·13-s + (−0.848 + 0.876i)15-s + (0.655 + 0.755i)16-s + 0.160i·17-s + (0.180 − 0.983i)18-s + 0.674i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.365 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.365 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.632871 + 0.928059i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.632871 + 0.928059i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.38 + 0.300i)T \) |
| 3 | \( 1 + (-1.24 - 1.20i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2.72iT - 5T^{2} \) |
| 11 | \( 1 + 2.04T + 11T^{2} \) |
| 13 | \( 1 - 4.44T + 13T^{2} \) |
| 17 | \( 1 - 0.660iT - 17T^{2} \) |
| 19 | \( 1 - 2.93iT - 19T^{2} \) |
| 23 | \( 1 + 1.04T + 23T^{2} \) |
| 29 | \( 1 - 2.06iT - 29T^{2} \) |
| 31 | \( 1 + 3.10iT - 31T^{2} \) |
| 37 | \( 1 + 9.52T + 37T^{2} \) |
| 41 | \( 1 + 7.85iT - 41T^{2} \) |
| 43 | \( 1 - 0.530iT - 43T^{2} \) |
| 47 | \( 1 + 1.04T + 47T^{2} \) |
| 53 | \( 1 - 10.5iT - 53T^{2} \) |
| 59 | \( 1 - 3.48T + 59T^{2} \) |
| 61 | \( 1 - 0.198T + 61T^{2} \) |
| 67 | \( 1 + 4.76iT - 67T^{2} \) |
| 71 | \( 1 - 5.52T + 71T^{2} \) |
| 73 | \( 1 - 3.03T + 73T^{2} \) |
| 79 | \( 1 - 10.9iT - 79T^{2} \) |
| 83 | \( 1 - 9.15T + 83T^{2} \) |
| 89 | \( 1 + 0.541iT - 89T^{2} \) |
| 97 | \( 1 - 6.91T + 97T^{2} \) |
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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
−10.60244670632663448577485863078, −10.30238856609001896423709729993, −9.237146622368610987473424669720, −8.417958800629306052075571654469, −7.67820939554400380627241153498, −6.75013796234021010429867922012, −5.65920955455416184518536111389, −3.82899864557663947176310052314, −3.11395638135808044408553156109, −2.00086475876803871196818615934,
0.802591594391509387108688329782, 1.94544828574442291092119880560, 3.35692249578913521259294234317, 5.01007849050164371499257189354, 6.13817092726198988569700770720, 7.06901515461337412849659038690, 8.115001319602612800914115161803, 8.567625466092219766690875032659, 9.201661075457129240377670875061, 10.15485447521119826375230525009