L(s) = 1 | + (0.227 + 1.71i)3-s − 5-s + (1.22 − 2.34i)7-s + (−2.89 + 0.781i)9-s + 4.73i·11-s + 5.25i·13-s + (−0.227 − 1.71i)15-s + 0.432·17-s − 6.30i·19-s + (4.30 + 1.56i)21-s + 0.332i·23-s + 25-s + (−2.00 − 4.79i)27-s + 7.48i·29-s − 0.0758i·31-s + ⋯ |
L(s) = 1 | + (0.131 + 0.991i)3-s − 0.447·5-s + (0.461 − 0.887i)7-s + (−0.965 + 0.260i)9-s + 1.42i·11-s + 1.45i·13-s + (−0.0587 − 0.443i)15-s + 0.104·17-s − 1.44i·19-s + (0.940 + 0.340i)21-s + 0.0692i·23-s + 0.200·25-s + (−0.385 − 0.922i)27-s + 1.39i·29-s − 0.0136i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 - 0.340i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.940 - 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.020510085\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.020510085\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.227 - 1.71i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-1.22 + 2.34i)T \) |
good | 11 | \( 1 - 4.73iT - 11T^{2} \) |
| 13 | \( 1 - 5.25iT - 13T^{2} \) |
| 17 | \( 1 - 0.432T + 17T^{2} \) |
| 19 | \( 1 + 6.30iT - 19T^{2} \) |
| 23 | \( 1 - 0.332iT - 23T^{2} \) |
| 29 | \( 1 - 7.48iT - 29T^{2} \) |
| 31 | \( 1 + 0.0758iT - 31T^{2} \) |
| 37 | \( 1 + 2.54T + 37T^{2} \) |
| 41 | \( 1 + 4.82T + 41T^{2} \) |
| 43 | \( 1 + 1.97T + 43T^{2} \) |
| 47 | \( 1 + 9.28T + 47T^{2} \) |
| 53 | \( 1 - 13.8iT - 53T^{2} \) |
| 59 | \( 1 - 5.65T + 59T^{2} \) |
| 61 | \( 1 - 5.11iT - 61T^{2} \) |
| 67 | \( 1 + 6.75T + 67T^{2} \) |
| 71 | \( 1 - 9.35iT - 71T^{2} \) |
| 73 | \( 1 - 2.75iT - 73T^{2} \) |
| 79 | \( 1 + 0.0508T + 79T^{2} \) |
| 83 | \( 1 + 4.12T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 11.1iT - 97T^{2} \) |
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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
−9.681073427112678324511000768100, −9.033809995815135909880113985956, −8.261983849394304333535643317807, −7.09265940338081971925027849683, −6.89900787833167970662461639421, −5.26437808316479532177151370006, −4.50044316363876591592150408755, −4.17408941689936786269544695498, −2.97104469621635394369939234042, −1.65537765008266962180932692575,
0.37512735807395934397342421468, 1.71455055750520950848123230528, 2.93091913396337149875108199362, 3.58175965924648052445001904539, 5.21547767975236880529945153848, 5.80019342548009042076950652348, 6.47032708700945611537928108276, 7.72865363574118504731980333953, 8.279783214277530368278232336873, 8.462686547140251538651341863250