| 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
−10.34365218162235631097624248652, −9.483300321896242361984723724428, −8.607924266435220238912916934369, −7.27422168757304524712798711403, −6.04030457170583891526095038169, −5.42086958051787372111031119237, −4.47770370171782596549223423339, −2.88007706346626999960985296408, −1.91040303412097135124768191667, −0.65994333186268255017729755101,
1.22100979329412891504131454139, 2.15832768320476541870184047582, 3.07284522074356298815473579569, 4.35541190180299316630970610837, 5.64626321858500896735946265724, 6.43880529005624259708785878301, 7.27602377260159353250934336692, 9.270674595186525853077785630837, 9.739338281072625947946129295140, 10.85857620934125001586007566866
