| L(s) = 1 | + (−22.0 − 15.6i)3-s − 161. i·5-s − 562.·7-s + (239. + 688. i)9-s + 1.45e3i·11-s + 1.09e3·13-s + (−2.52e3 + 3.55e3i)15-s + 4.05e3i·17-s + 1.57e3·19-s + (1.23e4 + 8.79e3i)21-s + 1.15e4i·23-s − 1.04e4·25-s + (5.51e3 − 1.88e4i)27-s − 3.72e4i·29-s + 2.34e3·31-s + ⋯ |
| L(s) = 1 | + (−0.814 − 0.579i)3-s − 1.29i·5-s − 1.63·7-s + (0.328 + 0.944i)9-s + 1.09i·11-s + 0.499·13-s + (−0.748 + 1.05i)15-s + 0.825i·17-s + 0.229·19-s + (1.33 + 0.949i)21-s + 0.950i·23-s − 0.667·25-s + (0.280 − 0.959i)27-s − 1.52i·29-s + 0.0786·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 + 0.579i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.814 + 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.8949035362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8949035362\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (22.0 + 15.6i)T \) |
| good | 5 | \( 1 + 161. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 562.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.45e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 1.09e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 4.05e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 1.57e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 1.15e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 3.72e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 2.34e3T + 8.87e8T^{2} \) |
| 37 | \( 1 + 6.80e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 3.60e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 1.01e5T + 6.32e9T^{2} \) |
| 47 | \( 1 - 5.03e3iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.09e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 2.50e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 3.18e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 2.26e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 2.35e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 6.94e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 2.05e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 1.26e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 1.03e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 3.17e5T + 8.32e11T^{2} \) |
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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
−11.66604390349120039016788731664, −10.21022968752168409262246859217, −9.507585344495163821020738216692, −8.290143940491428224305521469189, −7.04845895767488788253895746828, −6.11300620383793516836541545601, −5.11573758486999451970572591983, −3.81801835592475183588215929891, −1.86788037824566329547430233635, −0.59796451643262717421328537728,
0.51550696482954503041608181715, 3.03155945803578378540929520520, 3.57209326032990098942301284565, 5.38386109914076046101448249995, 6.54167840281144574858487955916, 6.82885180719450072499279682071, 8.784385467257626933202483284282, 9.865004893528586586965790640107, 10.57216477859651058858846350340, 11.30351736768351526924056609955
