| L(s) = 1 | + (−1.22 + 0.707i)2-s + (−0.866 + 1.5i)3-s + (0.999 − 1.73i)4-s − 2.44i·6-s + (−1.88 + 6.74i)7-s + 2.82i·8-s + (−1.5 − 2.59i)9-s + (−3 + 5.19i)11-s + (1.73 + 2.99i)12-s − 17.8·13-s + (−2.46 − 9.58i)14-s + (−2.00 − 3.46i)16-s + (−9.37 + 16.2i)17-s + (3.67 + 2.12i)18-s + (14.7 − 8.51i)19-s + ⋯ |
| L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.288 + 0.5i)3-s + (0.249 − 0.433i)4-s − 0.408i·6-s + (−0.268 + 0.963i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.272 + 0.472i)11-s + (0.144 + 0.249i)12-s − 1.37·13-s + (−0.176 − 0.684i)14-s + (−0.125 − 0.216i)16-s + (−0.551 + 0.955i)17-s + (0.204 + 0.117i)18-s + (0.775 − 0.447i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1552404828\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1552404828\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.22 - 0.707i)T \) |
| 3 | \( 1 + (0.866 - 1.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.88 - 6.74i)T \) |
| good | 11 | \( 1 + (3 - 5.19i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 17.8T + 169T^{2} \) |
| 17 | \( 1 + (9.37 - 16.2i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-14.7 + 8.51i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-11.6 + 6.72i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 33.9T + 841T^{2} \) |
| 31 | \( 1 + (-12.7 - 7.37i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (5.17 - 2.98i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 35.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 15.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-16.6 - 28.7i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-29.9 - 17.2i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (23.6 + 13.6i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (34.9 - 20.1i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (99.0 + 57.1i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 18.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-58.5 + 101. i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (44.1 + 76.5i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 75.7T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-18 + 10.3i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 30.5T + 9.40e3T^{2} \) |
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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
−9.349073563122696947534479091506, −9.045298410679327240287387468503, −7.894664032445322884653935038826, −7.10318628845363099335309184622, −6.14793707797283936256385238757, −5.30872022134781460642886449265, −4.55262277648723554803412645932, −3.02676227075413473795041790642, −1.97893248775238046285424303892, −0.07250510477860099331002743596,
0.966127967918405371616556982602, 2.36285830344715332842668071881, 3.38627102339797634936611315838, 4.65230134354830220123584549205, 5.65632137423846926537700894443, 6.94194143321855771628891148376, 7.32180446876295430904858997888, 8.090800043207294762443793209108, 9.269628081966063527213161121426, 9.854375093201783969197700340010
