L(s) = 1 | + (−0.559 + 0.968i)2-s + (0.913 + 0.406i)3-s + (−0.125 − 0.217i)4-s + (0.719 + 1.24i)5-s + (−0.904 + 0.657i)6-s − 0.837·8-s + (0.669 + 0.743i)9-s − 1.60·10-s + (0.882 − 1.52i)11-s + (−0.0262 − 0.249i)12-s + (0.150 + 1.43i)15-s + (0.593 − 1.02i)16-s − 1.87·17-s + (−1.09 + 0.232i)18-s + (0.180 − 0.312i)20-s + ⋯ |
L(s) = 1 | + (−0.559 + 0.968i)2-s + (0.913 + 0.406i)3-s + (−0.125 − 0.217i)4-s + (0.719 + 1.24i)5-s + (−0.904 + 0.657i)6-s − 0.837·8-s + (0.669 + 0.743i)9-s − 1.60·10-s + (0.882 − 1.52i)11-s + (−0.0262 − 0.249i)12-s + (0.150 + 1.43i)15-s + (0.593 − 1.02i)16-s − 1.87·17-s + (−1.09 + 0.232i)18-s + (0.180 − 0.312i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 - 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 - 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.351017016\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.351017016\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.913 - 0.406i)T \) |
| 239 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (0.559 - 0.968i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.719 - 1.24i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.882 + 1.52i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + 1.87T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.990 - 1.71i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.997 - 1.72i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - 0.618T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.615 + 1.06i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−9.178312158704276795696348795450, −8.823129678087060018343394305075, −8.223106376251016510978222542127, −7.08455430012716520501591862539, −6.70242280877291376818905449965, −6.08463188820226165367749785126, −4.94347831846104095830507813923, −3.47944618499770605831596990321, −3.11316743310051897318152575735, −1.96572288657831685437620622318,
1.05986225903764420764233468530, 2.12143919687652681176087957271, 2.30945806659477409260704465565, 4.05106093178787892070921336508, 4.50874685368903783486580084607, 5.93245664291958416237991128514, 6.61984516582613379738674943950, 7.63088433137655421991834393250, 8.614938489689597298905845780869, 9.074414344730194400395686495481