L(s) = 1 | + (1.23 + 0.430i)2-s + (0.997 − 0.0747i)3-s + (0.548 + 0.437i)4-s + (1.26 + 0.337i)6-s + (−0.206 − 0.329i)8-s + (0.988 − 0.149i)9-s + (0.579 + 0.395i)12-s + (−0.145 + 0.0332i)13-s + (−0.269 − 1.17i)16-s + (1.28 + 0.242i)18-s + (−0.433 + 0.900i)23-s + (−0.230 − 0.312i)24-s + (0.623 − 0.781i)25-s + (−0.193 − 0.0218i)26-s + (0.974 − 0.222i)27-s + ⋯ |
L(s) = 1 | + (1.23 + 0.430i)2-s + (0.997 − 0.0747i)3-s + (0.548 + 0.437i)4-s + (1.26 + 0.337i)6-s + (−0.206 − 0.329i)8-s + (0.988 − 0.149i)9-s + (0.579 + 0.395i)12-s + (−0.145 + 0.0332i)13-s + (−0.269 − 1.17i)16-s + (1.28 + 0.242i)18-s + (−0.433 + 0.900i)23-s + (−0.230 − 0.312i)24-s + (0.623 − 0.781i)25-s + (−0.193 − 0.0218i)26-s + (0.974 − 0.222i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.860047042\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.860047042\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.997 + 0.0747i)T \) |
| 23 | \( 1 + (0.433 - 0.900i)T \) |
| 29 | \( 1 + (0.930 - 0.365i)T \) |
good | 2 | \( 1 + (-1.23 - 0.430i)T + (0.781 + 0.623i)T^{2} \) |
| 5 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 7 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 11 | \( 1 + (0.433 + 0.900i)T^{2} \) |
| 13 | \( 1 + (0.145 - 0.0332i)T + (0.900 - 0.433i)T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + (0.974 - 0.222i)T^{2} \) |
| 31 | \( 1 + (0.638 - 1.82i)T + (-0.781 - 0.623i)T^{2} \) |
| 37 | \( 1 + (-0.433 + 0.900i)T^{2} \) |
| 41 | \( 1 + (-0.0528 + 0.0528i)T - iT^{2} \) |
| 43 | \( 1 + (-0.781 + 0.623i)T^{2} \) |
| 47 | \( 1 + (1.55 + 0.975i)T + (0.433 + 0.900i)T^{2} \) |
| 53 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 59 | \( 1 - 1.24iT - T^{2} \) |
| 61 | \( 1 + (-0.974 - 0.222i)T^{2} \) |
| 67 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 71 | \( 1 + (0.250 + 1.09i)T + (-0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (0.531 + 1.51i)T + (-0.781 + 0.623i)T^{2} \) |
| 79 | \( 1 + (0.433 - 0.900i)T^{2} \) |
| 83 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 89 | \( 1 + (-0.781 - 0.623i)T^{2} \) |
| 97 | \( 1 + (-0.974 + 0.222i)T^{2} \) |
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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
−9.260974523302931432286447719117, −8.583716221969390615497590646996, −7.58141320388723007632179119121, −7.00279711966458062147183987377, −6.19360820413855727697365459458, −5.20790759415756223285761063243, −4.50142429364711200693939310545, −3.57250708186717431058959886651, −2.99680803331520131282515602542, −1.68641105878167340453924294066,
1.86070081484915904155973427588, 2.66936380428746742247037722407, 3.55976508050854091353391775293, 4.21172041634797126770108589386, 5.01724888430076301095408770973, 5.93481860292203255969959580227, 6.89457137675734726230479079772, 7.84381153009720210691645935315, 8.493956709784587564398986571599, 9.358976496039190604821025802371