L(s) = 1 | + (−0.0607 − 1.99i)2-s + 5.37i·3-s + (−3.99 + 0.242i)4-s − 5.82·5-s + (10.7 − 0.326i)6-s + 5.45i·7-s + (0.728 + 7.96i)8-s − 19.9·9-s + (0.353 + 11.6i)10-s + 1.60i·11-s + (−1.30 − 21.4i)12-s + 23.5·13-s + (10.8 − 0.331i)14-s − 31.3i·15-s + (15.8 − 1.94i)16-s − 5.92·17-s + ⋯ |
L(s) = 1 | + (−0.0303 − 0.999i)2-s + 1.79i·3-s + (−0.998 + 0.0607i)4-s − 1.16·5-s + (1.79 − 0.0544i)6-s + 0.778i·7-s + (0.0910 + 0.995i)8-s − 2.21·9-s + (0.0353 + 1.16i)10-s + 0.146i·11-s + (−0.108 − 1.78i)12-s + 1.80·13-s + (0.778 − 0.0236i)14-s − 2.08i·15-s + (0.992 − 0.121i)16-s − 0.348·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0607 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0607 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.534833 + 0.568370i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.534833 + 0.568370i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.0607 + 1.99i)T \) |
| 19 | \( 1 + 4.35iT \) |
good | 3 | \( 1 - 5.37iT - 9T^{2} \) |
| 5 | \( 1 + 5.82T + 25T^{2} \) |
| 7 | \( 1 - 5.45iT - 49T^{2} \) |
| 11 | \( 1 - 1.60iT - 121T^{2} \) |
| 13 | \( 1 - 23.5T + 169T^{2} \) |
| 17 | \( 1 + 5.92T + 289T^{2} \) |
| 23 | \( 1 - 26.6iT - 529T^{2} \) |
| 29 | \( 1 + 1.49T + 841T^{2} \) |
| 31 | \( 1 - 31.3iT - 961T^{2} \) |
| 37 | \( 1 - 26.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 44.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 27.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 32.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 76.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + 33.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 53.0T + 3.72e3T^{2} \) |
| 67 | \( 1 - 76.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 59.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 49.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 23.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 137. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 116.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 65.7T + 9.40e3T^{2} \) |
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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
−14.79987553268109375800197592405, −13.39885866975245257235512754916, −11.74695678075890719116852727257, −11.29210774393287323246159421883, −10.27339782917484744590981006188, −9.026314259659546360153594681789, −8.402034392153080377471842206311, −5.48179916046875205790189474457, −4.15887494973096110744644782275, −3.32541785713982167546021892250,
0.72099987330905502382450209364, 3.90352909334339499873127467989, 6.08373680702418033094449666325, 7.00524092322590044734123891049, 7.960146630577004036951313767473, 8.602233552729455584144185441316, 10.92458425363601150563000757387, 12.09841502768406806587730542996, 13.25400723609121671165070542546, 13.73286683528631366153765071728