| L(s) = 1 | − 4i·2-s − 9i·3-s − 16·4-s − 36·6-s − 176i·7-s + 64i·8-s − 81·9-s − 60·11-s + 144i·12-s − 658i·13-s − 704·14-s + 256·16-s + 414i·17-s + 324i·18-s − 956·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s − 1.35i·7-s + 0.353i·8-s − 0.333·9-s − 0.149·11-s + 0.288i·12-s − 1.07i·13-s − 0.959·14-s + 0.250·16-s + 0.347i·17-s + 0.235i·18-s − 0.607·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6988733567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6988733567\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4iT \) |
| 3 | \( 1 + 9iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 176iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 60T + 1.61e5T^{2} \) |
| 13 | \( 1 + 658iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 414iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 956T + 2.47e6T^{2} \) |
| 23 | \( 1 - 600iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 5.57e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.45e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.91e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.33e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.96e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 3.12e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.63e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.10e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.68e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 6.12e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.55e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 7.44e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.46e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 3.27e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.66e5iT - 8.58e9T^{2} \) |
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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
−11.23891891189954213979634646636, −10.63116002478468723701034913328, −9.575084149249534643497780426903, −8.141401536547654964499188224041, −7.34167751211596913880753574596, −5.90207857256970746142712854777, −4.37200567852592493197811333331, −3.11671009215942532711482211859, −1.47035016809770809071589062797, −0.23627563398808766460666185056,
2.28695729710342799180328062468, 4.01463765050741375819661339958, 5.27540039637735631640722986797, 6.17061270592848741183560595170, 7.52992944312559253290244154219, 8.985090491676490937216647207068, 9.195786665547226751348102148403, 10.72053030259340718593677856013, 11.83128015831663460529586526303, 12.78000716418146201544645701105
