L(s) = 1 | − 15.2i·2-s + (−26.4 + 5.62i)3-s − 167.·4-s − 4.50i·5-s + (85.5 + 401. i)6-s + 443.·7-s + 1.56e3i·8-s + (665. − 297. i)9-s − 68.4·10-s + 1.98e3i·11-s + (4.41e3 − 940. i)12-s − 3.41e3·13-s − 6.74e3i·14-s + (25.3 + 118. i)15-s + 1.31e4·16-s − 4.05e3i·17-s + ⋯ |
L(s) = 1 | − 1.90i·2-s + (−0.978 + 0.208i)3-s − 2.61·4-s − 0.0360i·5-s + (0.395 + 1.85i)6-s + 1.29·7-s + 3.06i·8-s + (0.913 − 0.407i)9-s − 0.0684·10-s + 1.49i·11-s + (2.55 − 0.544i)12-s − 1.55·13-s − 2.45i·14-s + (0.00750 + 0.0352i)15-s + 3.21·16-s − 0.825i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.208 + 0.978i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.208 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.823620 - 0.666668i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.823620 - 0.666668i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (26.4 - 5.62i)T \) |
| 23 | \( 1 - 2.53e3iT \) |
good | 2 | \( 1 + 15.2iT - 64T^{2} \) |
| 5 | \( 1 + 4.50iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 443.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.98e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 3.41e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 4.05e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 1.55e3T + 4.70e7T^{2} \) |
| 29 | \( 1 - 2.94e3iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 5.14e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 2.69e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 2.84e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 4.97e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.26e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.56e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 1.43e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 2.71e4T + 5.15e10T^{2} \) |
| 67 | \( 1 - 8.08e4T + 9.04e10T^{2} \) |
| 71 | \( 1 + 5.83e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 5.94e4T + 1.51e11T^{2} \) |
| 79 | \( 1 - 1.28e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 1.06e6iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 5.95e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 1.21e6T + 8.32e11T^{2} \) |
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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
−12.53119096419385249405367643108, −12.06485521641288543839645091612, −11.17946155642148065471313476094, −10.16508488218218053290147620190, −9.387572171906062077636256381407, −7.54256222662041236630340748276, −4.86338143670471688062928508589, −4.64309555383169634542396063525, −2.36690937734265361248699724651, −1.00846368737762621801378012903,
0.65679066795612415766082471650, 4.56065929275047006486826369877, 5.39806907087534029540889266827, 6.49597012840150128834765822931, 7.68352721598220947750862373278, 8.532237461067447479341454044322, 10.19701583313281314976666400232, 11.62941190224967099309223037723, 12.99567635682780321433735864289, 14.15543353436729223338682027640