| L(s) = 1 | + (8 + 13.8i)3-s + (−8 + 13.8i)5-s + (−6.49 + 11.2i)9-s + (38 + 65.8i)11-s + 880·13-s − 255.·15-s + (528 + 914. i)17-s + (−968 + 1.67e3i)19-s + (−468 + 810. i)23-s + (1.43e3 + 2.48e3i)25-s + 3.68e3·27-s − 3.98e3·29-s + (−784 − 1.35e3i)31-s + (−607. + 1.05e3i)33-s + (−2.46e3 + 4.27e3i)37-s + ⋯ |
| L(s) = 1 | + (0.513 + 0.888i)3-s + (−0.143 + 0.247i)5-s + (−0.0267 + 0.0463i)9-s + (0.0946 + 0.164i)11-s + 1.44·13-s − 0.293·15-s + (0.443 + 0.767i)17-s + (−0.615 + 1.06i)19-s + (−0.184 + 0.319i)23-s + (0.459 + 0.795i)25-s + 0.971·27-s − 0.879·29-s + (−0.146 − 0.253i)31-s + (−0.0971 + 0.168i)33-s + (−0.296 + 0.513i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.223325416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.223325416\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (-8 - 13.8i)T + (-121.5 + 210. i)T^{2} \) |
| 5 | \( 1 + (8 - 13.8i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-38 - 65.8i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 - 880T + 3.71e5T^{2} \) |
| 17 | \( 1 + (-528 - 914. i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (968 - 1.67e3i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (468 - 810. i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + 3.98e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (784 + 1.35e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (2.46e3 - 4.27e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + 1.58e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.64e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.03e4 + 1.79e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-1.87e4 - 3.23e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (1.05e4 + 1.83e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-1.49e3 + 2.59e3i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-2.29e4 - 3.96e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 4.98e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (-2.81e4 - 4.87e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (2.03e4 - 3.52e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 1.12e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + (3.21e4 - 5.56e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + 2.27e3T + 8.58e9T^{2} \) |
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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
−11.81561721452964933516469377766, −10.69561159184162142957522118339, −10.03886602568073940224929296641, −8.913900183961893331339833135641, −8.169400650125239181379762570001, −6.73325904190581035123164386360, −5.55214891417132274776228048930, −3.97068208298486547358754977853, −3.43351682167006167427289163902, −1.53492376867214420360106214468,
0.66966273850264665563748043896, 1.95317081922379601423345697329, 3.34176579791895165297360408445, 4.83078896996562077526495690341, 6.30201142062489954899870952425, 7.23631401458945937679305316037, 8.337777881192578572905154500694, 8.942898202073100229609197807777, 10.41391690129658148311588028599, 11.40397241812851182311984451600
