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 + (6.99 + 0.264i)7-s − 2.82·8-s + (1.5 − 2.59i)9-s + (2.03 + 3.52i)11-s + (−2.99 − 1.73i)12-s − 11.1i·13-s + (5.26 − 8.38i)14-s + (−2.00 + 3.46i)16-s + (−2.50 + 1.44i)17-s + (−2.12 − 3.67i)18-s + (17.9 + 10.3i)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 + (0.999 + 0.0377i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (0.184 + 0.320i)11-s + (−0.249 − 0.144i)12-s − 0.855i·13-s + (0.376 − 0.598i)14-s + (−0.125 + 0.216i)16-s + (−0.147 + 0.0851i)17-s + (−0.117 − 0.204i)18-s + (0.944 + 0.545i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.163 + 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.103980170\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.103980170\) |
\(L(2)\) |
|
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 | \( 1 + (-6.99 - 0.264i)T \) |
good | 11 | \( 1 + (-2.03 - 3.52i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 11.1iT - 169T^{2} \) |
| 17 | \( 1 + (2.50 - 1.44i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-17.9 - 10.3i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-15.3 + 26.5i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 5.75T + 841T^{2} \) |
| 31 | \( 1 + (-3.32 + 1.91i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-4.46 + 7.72i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 56.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 7.16T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-41.2 - 23.8i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (16.6 + 28.8i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (38.3 - 22.1i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (60.3 + 34.8i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-46.0 - 79.7i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 6.64T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-45.3 + 26.2i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-26.7 + 46.2i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 116. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (85.6 + 49.4i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 132. iT - 9.40e3T^{2} \) |
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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
−9.504618875174689079498605393071, −8.692956390872890475131611922855, −7.904438299109186541561885730289, −7.12055737786394976446327840995, −5.88373974786232136719397822594, −5.01608596032699837813720482729, −4.09151944290980217863179167621, −3.00965999711471516929224319248, −2.00853254797421447498372942353, −0.896201319176379849006351668015,
1.38998482378959138138130477649, 2.80815435311863546691555309124, 3.90294474550199845204217500029, 4.77379848144166065716920238625, 5.49496234359285937564923531606, 6.68802099996647844141075419490, 7.50329538338742957881293606402, 8.178400742388003151026618564923, 9.106696737590100738939857050092, 9.576604318400768773592973588104