| L(s) = 1 | + (−11.0 − 11.0i)3-s + (55.7 + 55.7i)5-s − 496.·7-s + 242. i·9-s + (694. − 694. i)11-s + (−214. + 214. i)13-s − 1.22e3i·15-s − 4.13e3·17-s + (−7.22e3 − 7.22e3i)19-s + (5.47e3 + 5.47e3i)21-s + 2.02e4·23-s − 9.40e3i·25-s + (2.67e3 − 2.67e3i)27-s + (−2.57e4 + 2.57e4i)29-s − 3.55e4i·31-s + ⋯ |
| L(s) = 1 | + (−0.408 − 0.408i)3-s + (0.446 + 0.446i)5-s − 1.44·7-s + 0.333i·9-s + (0.521 − 0.521i)11-s + (−0.0976 + 0.0976i)13-s − 0.364i·15-s − 0.840·17-s + (−1.05 − 1.05i)19-s + (0.591 + 0.591i)21-s + 1.66·23-s − 0.601i·25-s + (0.136 − 0.136i)27-s + (−1.05 + 1.05i)29-s − 1.19i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 - 0.727i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.686 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.9866581093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9866581093\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (11.0 + 11.0i)T \) |
| good | 5 | \( 1 + (-55.7 - 55.7i)T + 1.56e4iT^{2} \) |
| 7 | \( 1 + 496.T + 1.17e5T^{2} \) |
| 11 | \( 1 + (-694. + 694. i)T - 1.77e6iT^{2} \) |
| 13 | \( 1 + (214. - 214. i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 + 4.13e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + (7.22e3 + 7.22e3i)T + 4.70e7iT^{2} \) |
| 23 | \( 1 - 2.02e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + (2.57e4 - 2.57e4i)T - 5.94e8iT^{2} \) |
| 31 | \( 1 + 3.55e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + (-1.20e4 - 1.20e4i)T + 2.56e9iT^{2} \) |
| 41 | \( 1 + 5.59e3iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (1.22e4 - 1.22e4i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 - 1.26e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + (-6.91e4 - 6.91e4i)T + 2.21e10iT^{2} \) |
| 59 | \( 1 + (1.01e5 - 1.01e5i)T - 4.21e10iT^{2} \) |
| 61 | \( 1 + (1.23e5 - 1.23e5i)T - 5.15e10iT^{2} \) |
| 67 | \( 1 + (3.94e5 + 3.94e5i)T + 9.04e10iT^{2} \) |
| 71 | \( 1 - 2.64e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 2.30e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 2.78e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (-6.05e5 - 6.05e5i)T + 3.26e11iT^{2} \) |
| 89 | \( 1 - 5.26e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 8.40e5T + 8.32e11T^{2} \) |
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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
−10.66333348579679780709948888634, −9.397370665624469640310775531030, −8.907953743079216936590898705333, −7.30991109899082659424118469557, −6.50520148240895790417955333315, −6.07987453929286637937675432472, −4.62204190134869735663949918961, −3.24998386919307551161540232825, −2.29766910727860829199745853129, −0.71511873711266588861105921403,
0.33470652535523211750301388670, 1.81836851906151536066339570511, 3.27614778681165678395109937178, 4.29233563709652762220754368981, 5.44779771356611449147319434779, 6.37658149237989912602028696624, 7.10462197298035166854482623588, 8.730090195429997751291325895120, 9.386173256435144239128462197142, 10.09283838188219540603645368538
