| L(s) = 1 | + (39.1 − 22.6i)2-s + (1.02e3 − 1.77e3i)4-s + (2.34e4 + 1.35e4i)5-s + (−1.92e4 − 3.33e4i)7-s − 9.26e4i·8-s + 1.22e6·10-s + (9.15e5 − 5.28e5i)11-s + (−3.18e6 + 5.51e6i)13-s + (−1.50e6 − 8.71e5i)14-s + (−2.09e6 − 3.63e6i)16-s + 2.54e6i·17-s + 1.58e6·19-s + (4.79e7 − 2.77e7i)20-s + (2.39e7 − 4.14e7i)22-s + (1.09e8 + 6.34e7i)23-s + ⋯ |
| L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (1.49 + 0.865i)5-s + (−0.163 − 0.283i)7-s − 0.353i·8-s + 1.22·10-s + (0.516 − 0.298i)11-s + (−0.659 + 1.14i)13-s + (−0.200 − 0.115i)14-s + (−0.125 − 0.216i)16-s + 0.105i·17-s + 0.0336·19-s + (0.749 − 0.432i)20-s + (0.210 − 0.365i)22-s + (0.742 + 0.428i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(4.442967661\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.442967661\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-39.1 + 22.6i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-2.34e4 - 1.35e4i)T + (1.22e8 + 2.11e8i)T^{2} \) |
| 7 | \( 1 + (1.92e4 + 3.33e4i)T + (-6.92e9 + 1.19e10i)T^{2} \) |
| 11 | \( 1 + (-9.15e5 + 5.28e5i)T + (1.56e12 - 2.71e12i)T^{2} \) |
| 13 | \( 1 + (3.18e6 - 5.51e6i)T + (-1.16e13 - 2.01e13i)T^{2} \) |
| 17 | \( 1 - 2.54e6iT - 5.82e14T^{2} \) |
| 19 | \( 1 - 1.58e6T + 2.21e15T^{2} \) |
| 23 | \( 1 + (-1.09e8 - 6.34e7i)T + (1.09e16 + 1.89e16i)T^{2} \) |
| 29 | \( 1 + (7.45e8 - 4.30e8i)T + (1.76e17 - 3.06e17i)T^{2} \) |
| 31 | \( 1 + (-3.60e8 + 6.25e8i)T + (-3.93e17 - 6.82e17i)T^{2} \) |
| 37 | \( 1 - 4.10e9T + 6.58e18T^{2} \) |
| 41 | \( 1 + (1.46e9 + 8.47e8i)T + (1.12e19 + 1.95e19i)T^{2} \) |
| 43 | \( 1 + (-5.93e9 - 1.02e10i)T + (-1.99e19 + 3.46e19i)T^{2} \) |
| 47 | \( 1 + (7.65e9 - 4.42e9i)T + (5.80e19 - 1.00e20i)T^{2} \) |
| 53 | \( 1 - 1.49e10iT - 4.91e20T^{2} \) |
| 59 | \( 1 + (-5.34e10 - 3.08e10i)T + (8.89e20 + 1.54e21i)T^{2} \) |
| 61 | \( 1 + (3.07e10 + 5.31e10i)T + (-1.32e21 + 2.29e21i)T^{2} \) |
| 67 | \( 1 + (4.81e10 - 8.34e10i)T + (-4.09e21 - 7.08e21i)T^{2} \) |
| 71 | \( 1 + 1.92e11iT - 1.64e22T^{2} \) |
| 73 | \( 1 + 2.52e11T + 2.29e22T^{2} \) |
| 79 | \( 1 + (-1.32e11 - 2.29e11i)T + (-2.95e22 + 5.11e22i)T^{2} \) |
| 83 | \( 1 + (5.02e11 - 2.89e11i)T + (5.34e22 - 9.25e22i)T^{2} \) |
| 89 | \( 1 - 2.67e11iT - 2.46e23T^{2} \) |
| 97 | \( 1 + (-7.00e11 - 1.21e12i)T + (-3.46e23 + 6.00e23i)T^{2} \) |
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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
−10.85857620934125001586007566866, −9.739338281072625947946129295140, −9.270674595186525853077785630837, −7.27602377260159353250934336692, −6.43880529005624259708785878301, −5.64626321858500896735946265724, −4.35541190180299316630970610837, −3.07284522074356298815473579569, −2.15832768320476541870184047582, −1.22100979329412891504131454139,
0.65994333186268255017729755101, 1.91040303412097135124768191667, 2.88007706346626999960985296408, 4.47770370171782596549223423339, 5.42086958051787372111031119237, 6.04030457170583891526095038169, 7.27422168757304524712798711403, 8.607924266435220238912916934369, 9.483300321896242361984723724428, 10.34365218162235631097624248652
