| L(s) = 1 | + (0.707 + 1.22i)2-s + (−1.5 − 0.866i)3-s + (−0.999 + 1.73i)4-s − 2.44i·6-s + (6.74 + 1.88i)7-s − 2.82·8-s + (1.5 + 2.59i)9-s + (−3 + 5.19i)11-s + (2.99 − 1.73i)12-s − 17.8i·13-s + (2.46 + 9.58i)14-s + (−2.00 − 3.46i)16-s + (16.2 + 9.37i)17-s + (−2.12 + 3.67i)18-s + (−14.7 + 8.51i)19-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.5 − 0.288i)3-s + (−0.249 + 0.433i)4-s − 0.408i·6-s + (0.963 + 0.268i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (−0.272 + 0.472i)11-s + (0.249 − 0.144i)12-s − 1.37i·13-s + (0.176 + 0.684i)14-s + (−0.125 − 0.216i)16-s + (0.955 + 0.551i)17-s + (−0.117 + 0.204i)18-s + (−0.775 + 0.447i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.995883210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.995883210\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.707 - 1.22i)T \) |
| 3 | \( 1 + (1.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-6.74 - 1.88i)T \) |
| good | 11 | \( 1 + (3 - 5.19i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 17.8iT - 169T^{2} \) |
| 17 | \( 1 + (-16.2 - 9.37i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (14.7 - 8.51i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-6.72 - 11.6i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 33.9T + 841T^{2} \) |
| 31 | \( 1 + (-12.7 - 7.37i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-2.98 - 5.17i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 35.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 15.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-28.7 + 16.6i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (17.2 - 29.9i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 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 | \( 1 + (57.1 - 99.0i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 18.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-101. - 58.5i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-44.1 - 76.5i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 75.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (18 - 10.3i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 30.5iT - 9.40e3T^{2} \) |
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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
−10.14802863769007090961007697213, −8.742170038129139658148715868850, −8.016932193942422549773717379970, −7.51361220962386110910883472952, −6.41103152225436121972839590276, −5.52528098995266827957616892796, −5.03333917412405149341693951036, −3.90660336074076932526908622176, −2.56368539806353365034816642355, −1.11493823110572250908218057583,
0.70526034254819769807356385531, 1.95328285233156467160214126113, 3.22865684946373180334805760597, 4.52233213244292252338028260725, 4.79134472636754897611101344435, 5.99237458461935296650770489821, 6.83238694909829739699430022500, 7.972009757025980291318337293798, 8.850771000464334723673956629708, 9.709456718186825294513546582969
