| L(s) = 1 | + (19.7 + 60.8i)2-s + (−2.00e3 − 1.45e3i)3-s + (−3.31e3 + 2.40e3i)4-s + (1.25e4 − 3.87e4i)5-s + (4.89e4 − 1.50e5i)6-s + (3.45e5 − 2.51e5i)7-s + (−2.12e5 − 1.54e5i)8-s + (1.40e6 + 4.31e6i)9-s + 2.60e6·10-s + (1.34e6 − 5.71e6i)11-s + 1.01e7·12-s + (−7.02e6 − 2.16e7i)13-s + (2.21e7 + 1.60e7i)14-s + (−8.16e7 + 5.92e7i)15-s + (5.18e6 − 1.59e7i)16-s + (2.72e7 − 8.39e7i)17-s + ⋯ |
| L(s) = 1 | + (0.218 + 0.672i)2-s + (−1.58 − 1.15i)3-s + (−0.404 + 0.293i)4-s + (0.360 − 1.10i)5-s + (0.428 − 1.31i)6-s + (1.11 − 0.807i)7-s + (−0.286 − 0.207i)8-s + (0.879 + 2.70i)9-s + 0.824·10-s + (0.228 − 0.973i)11-s + 0.980·12-s + (−0.403 − 1.24i)13-s + (0.785 + 0.570i)14-s + (−1.84 + 1.34i)15-s + (0.0772 − 0.237i)16-s + (0.274 − 0.843i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 + 0.344i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.938 + 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.041183669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.041183669\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-19.7 - 60.8i)T \) |
| 11 | \( 1 + (-1.34e6 + 5.71e6i)T \) |
| good | 3 | \( 1 + (2.00e3 + 1.45e3i)T + (4.92e5 + 1.51e6i)T^{2} \) |
| 5 | \( 1 + (-1.25e4 + 3.87e4i)T + (-9.87e8 - 7.17e8i)T^{2} \) |
| 7 | \( 1 + (-3.45e5 + 2.51e5i)T + (2.99e10 - 9.21e10i)T^{2} \) |
| 13 | \( 1 + (7.02e6 + 2.16e7i)T + (-2.45e14 + 1.78e14i)T^{2} \) |
| 17 | \( 1 + (-2.72e7 + 8.39e7i)T + (-8.01e15 - 5.82e15i)T^{2} \) |
| 19 | \( 1 + (2.50e7 + 1.81e7i)T + (1.29e16 + 3.99e16i)T^{2} \) |
| 23 | \( 1 - 5.27e7T + 5.04e17T^{2} \) |
| 29 | \( 1 + (2.14e9 - 1.55e9i)T + (3.17e18 - 9.75e18i)T^{2} \) |
| 31 | \( 1 + (-9.62e8 - 2.96e9i)T + (-1.97e19 + 1.43e19i)T^{2} \) |
| 37 | \( 1 + (1.13e10 - 8.24e9i)T + (7.52e19 - 2.31e20i)T^{2} \) |
| 41 | \( 1 + (3.60e10 + 2.61e10i)T + (2.85e20 + 8.79e20i)T^{2} \) |
| 43 | \( 1 - 1.26e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (-3.13e10 - 2.27e10i)T + (1.68e21 + 5.19e21i)T^{2} \) |
| 53 | \( 1 + (-1.78e10 - 5.50e10i)T + (-2.10e22 + 1.53e22i)T^{2} \) |
| 59 | \( 1 + (1.35e10 - 9.84e9i)T + (3.24e22 - 9.98e22i)T^{2} \) |
| 61 | \( 1 + (-2.64e10 + 8.14e10i)T + (-1.30e23 - 9.51e22i)T^{2} \) |
| 67 | \( 1 - 9.59e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + (5.27e11 - 1.62e12i)T + (-9.42e23 - 6.84e23i)T^{2} \) |
| 73 | \( 1 + (-1.60e11 + 1.16e11i)T + (5.16e23 - 1.59e24i)T^{2} \) |
| 79 | \( 1 + (1.83e11 + 5.65e11i)T + (-3.77e24 + 2.74e24i)T^{2} \) |
| 83 | \( 1 + (-9.56e11 + 2.94e12i)T + (-7.17e24 - 5.21e24i)T^{2} \) |
| 89 | \( 1 - 4.95e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (9.19e11 + 2.83e12i)T + (-5.44e25 + 3.95e25i)T^{2} \) |
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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
−13.92811856164771704969678036855, −13.04045459706942256082312901030, −11.95658112986230810785348958249, −10.69125434822083288527256773669, −8.243033709129985182076212293801, −7.14356179176570269098586167480, −5.53778521187223442058445302648, −4.94576798509122167508445960167, −1.26489398679333500215241325243, −0.46082763117924982635207621458,
1.87677862248426831569472707069, 4.08352124355517341760840927207, 5.22834561969843773015447826807, 6.53626640892313210331605192241, 9.448642230924297965998327739472, 10.48051317366494611470046602319, 11.44543679558620676514209825011, 12.16840144334102177740272962701, 14.62282295549626364109784160457, 15.16814873880274590650870867238
