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 + (6.34 + 2.94i)7-s + 2.82i·8-s + (−1.5 − 2.59i)9-s + (−8.87 + 15.3i)11-s + (1.73 + 2.99i)12-s − 8.10·13-s + (−9.86 + 0.879i)14-s + (−2.00 − 3.46i)16-s + (3.93 − 6.81i)17-s + (3.67 + 2.12i)18-s + (−17.5 + 10.1i)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 + (0.907 + 0.421i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.806 + 1.39i)11-s + (0.144 + 0.249i)12-s − 0.623·13-s + (−0.704 + 0.0628i)14-s + (−0.125 − 0.216i)16-s + (0.231 − 0.401i)17-s + (0.204 + 0.117i)18-s + (−0.926 + 0.534i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.697 + 0.716i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.697 + 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2831412654\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2831412654\) |
\(L(2)\) |
|
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 | \( 1 + (-6.34 - 2.94i)T \) |
good | 11 | \( 1 + (8.87 - 15.3i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 8.10T + 169T^{2} \) |
| 17 | \( 1 + (-3.93 + 6.81i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (17.5 - 10.1i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-28.2 + 16.2i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 25.0T + 841T^{2} \) |
| 31 | \( 1 + (46.0 + 26.5i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (37.7 - 21.7i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 70.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 2.43iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (7.69 + 13.3i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-44.0 - 25.4i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (22.9 + 13.2i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (28.7 - 16.6i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (86.4 + 49.9i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 97.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-27.0 + 46.7i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (37.3 + 64.7i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 85.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-115. + 66.5i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 49.9T + 9.40e3T^{2} \) |
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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
−10.27062178645715798560030154472, −9.397308328859369278259418496353, −8.637236600252261217216948717814, −7.74172339181469022039871124683, −7.09351243257472087442742504815, −5.97386975661844773710626003675, −4.93090675831182888536939921605, −4.58328623014867699728725647040, −2.75942513541715793545118093533, −1.69601663446159696786648794987,
0.11426716780772280944379130882, 1.28729406981344490812100298869, 2.46462718822651232575445590277, 3.60261699769300254521581203692, 4.97187140543709696773738636806, 5.71687564457689686338540286544, 7.01910767209600825594035387248, 7.51442274732145079970431686854, 8.546448394291162208639872444296, 8.882641382092617791858041801823