L(s) = 1 | + i·2-s + 3-s − 4-s + (−0.866 + 0.5i)5-s + i·6-s − i·8-s + 9-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s − 12-s + (0.866 + 0.5i)13-s + (−0.866 + 0.5i)15-s + 16-s + (−0.5 − 0.866i)17-s + i·18-s + (0.5 − 0.866i)19-s + ⋯ |
L(s) = 1 | + i·2-s + 3-s − 4-s + (−0.866 + 0.5i)5-s + i·6-s − i·8-s + 9-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s − 12-s + (0.866 + 0.5i)13-s + (−0.866 + 0.5i)15-s + 16-s + (−0.5 − 0.866i)17-s + i·18-s + (0.5 − 0.866i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.544606045\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.544606045\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 - 2iT - T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−8.761488878752206019721519579212, −8.150329368569314368903196362278, −7.36036724282168723739839618641, −6.83811171507301069249461138778, −6.22507845008048094912120490269, −4.95454209460848301146379348683, −4.25736269662469275128700646031, −3.47396956530080476909438730049, −2.83524444749774969684142421509, −1.05296749719297495937178609394,
1.21448026492260433858977212928, 1.99308451640444034374467753981, 3.29448408373054880228499096418, 3.71553242211967050102927765429, 4.44605186805871331387859412872, 5.21814877617517081970210355329, 6.48992541435557533651385408336, 7.51822635027061927067082922851, 8.122373309042254827113691193040, 8.755458906234641627905774061535