L(s) = 1 | + (−0.809 − 0.587i)4-s + (0.309 + 0.951i)5-s + 0.618·7-s + (0.309 + 0.951i)9-s + (0.190 − 0.587i)11-s + (0.309 + 0.951i)16-s + (1.30 − 0.951i)17-s + (−0.809 + 0.587i)19-s + (0.309 − 0.951i)20-s + (0.190 − 0.587i)23-s + (−0.809 + 0.587i)25-s + (−0.500 − 0.363i)28-s + (0.190 + 0.587i)35-s + (0.309 − 0.951i)36-s − 1.61·43-s + (−0.5 + 0.363i)44-s + ⋯ |
L(s) = 1 | + (−0.809 − 0.587i)4-s + (0.309 + 0.951i)5-s + 0.618·7-s + (0.309 + 0.951i)9-s + (0.190 − 0.587i)11-s + (0.309 + 0.951i)16-s + (1.30 − 0.951i)17-s + (−0.809 + 0.587i)19-s + (0.309 − 0.951i)20-s + (0.190 − 0.587i)23-s + (−0.809 + 0.587i)25-s + (−0.500 − 0.363i)28-s + (0.190 + 0.587i)35-s + (0.309 − 0.951i)36-s − 1.61·43-s + (−0.5 + 0.363i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8385586567\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8385586567\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.309 - 0.951i)T \) |
| 19 | \( 1 + (0.809 - 0.587i)T \) |
good | 2 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 3 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 - 0.618T + T^{2} \) |
| 11 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + 1.61T + T^{2} \) |
| 47 | \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
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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
−11.01240611397593807139001346037, −10.34057463346922722045445371387, −9.726106130176514588999452767824, −8.501925301299130963833091403594, −7.75199927279110144898271175156, −6.55261603617343811591781833819, −5.51754326709316008496193761884, −4.71224558264376277973019136988, −3.34161278065914401490037529893, −1.77277782451024716362909059922,
1.46582343848180890479956299178, 3.50517554469108556152813166591, 4.47260810060413977729218084541, 5.27836402058779652395305510662, 6.55089706267017693924717340692, 7.888705089503700860205246866215, 8.452419387931001697072544752155, 9.448762793123509315103132686116, 9.910558876033988650893968898905, 11.37886454512092168024259263242