L(s) = 1 | + (−2.23 − 0.0113i)5-s + (−0.471 − 0.471i)7-s − 0.335·11-s + (3.50 + 3.50i)13-s + (2.53 − 2.53i)17-s − 4.07·19-s + (−6.20 − 6.20i)23-s + (4.99 + 0.0506i)25-s + 2.42i·29-s + 6.41·31-s + (1.04 + 1.06i)35-s + (2.24 − 2.24i)37-s − 5.80i·41-s + (−4.87 − 4.87i)43-s + (1.68 − 1.68i)47-s + ⋯ |
L(s) = 1 | + (−0.999 − 0.00506i)5-s + (−0.178 − 0.178i)7-s − 0.101·11-s + (0.971 + 0.971i)13-s + (0.614 − 0.614i)17-s − 0.934·19-s + (−1.29 − 1.29i)23-s + (0.999 + 0.0101i)25-s + 0.450i·29-s + 1.15·31-s + (0.177 + 0.179i)35-s + (0.369 − 0.369i)37-s − 0.906i·41-s + (−0.743 − 0.743i)43-s + (0.245 − 0.245i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.160 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.005391929\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.005391929\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.23 + 0.0113i)T \) |
good | 7 | \( 1 + (0.471 + 0.471i)T + 7iT^{2} \) |
| 11 | \( 1 + 0.335T + 11T^{2} \) |
| 13 | \( 1 + (-3.50 - 3.50i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.53 + 2.53i)T - 17iT^{2} \) |
| 19 | \( 1 + 4.07T + 19T^{2} \) |
| 23 | \( 1 + (6.20 + 6.20i)T + 23iT^{2} \) |
| 29 | \( 1 - 2.42iT - 29T^{2} \) |
| 31 | \( 1 - 6.41T + 31T^{2} \) |
| 37 | \( 1 + (-2.24 + 2.24i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.80iT - 41T^{2} \) |
| 43 | \( 1 + (4.87 + 4.87i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.68 + 1.68i)T - 47iT^{2} \) |
| 53 | \( 1 + (3.05 - 3.05i)T - 53iT^{2} \) |
| 59 | \( 1 + 12.2iT - 59T^{2} \) |
| 61 | \( 1 + 7.49iT - 61T^{2} \) |
| 67 | \( 1 + (-5.55 + 5.55i)T - 67iT^{2} \) |
| 71 | \( 1 + 13.4iT - 71T^{2} \) |
| 73 | \( 1 + (-5.05 + 5.05i)T - 73iT^{2} \) |
| 79 | \( 1 - 8.85iT - 79T^{2} \) |
| 83 | \( 1 + (4.78 - 4.78i)T - 83iT^{2} \) |
| 89 | \( 1 + 8.33T + 89T^{2} \) |
| 97 | \( 1 + (-10.1 - 10.1i)T + 97iT^{2} \) |
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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
−9.232866858607842570796637576747, −8.422824823491591466799477151658, −7.912586132925192981170988046449, −6.79220243318536773140848994862, −6.33446429601450101712126379005, −4.99225147487838061803241043393, −4.13420748332200662622822696348, −3.45445027151846469344379254338, −2.09837761637254060865928909933, −0.45981800609566973412716776244,
1.19516172169149738676647004281, 2.83285329795806693875423564151, 3.71287663163745686876681818488, 4.47851020570827060555430167421, 5.75057639534687533834803426961, 6.30310682217363845448230241319, 7.52637821525719825345737379399, 8.140746184671750228161774090769, 8.616851460532416321471965475497, 9.874734851346607710471506831152