L(s) = 1 | − 0.618i·2-s + 0.618·4-s − i·8-s − 9-s + 0.618·11-s + 0.618i·13-s + 0.618i·18-s + 1.61·19-s − 0.381i·22-s − 1.61i·23-s + 0.381·26-s + 0.618·31-s − 0.999i·32-s − 0.618·36-s − 1.00i·38-s + ⋯ |
L(s) = 1 | − 0.618i·2-s + 0.618·4-s − i·8-s − 9-s + 0.618·11-s + 0.618i·13-s + 0.618i·18-s + 1.61·19-s − 0.381i·22-s − 1.61i·23-s + 0.381·26-s + 0.618·31-s − 0.999i·32-s − 0.618·36-s − 1.00i·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.388118062\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.388118062\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 79 | \( 1 + T \) |
good | 2 | \( 1 + 0.618iT - T^{2} \) |
| 3 | \( 1 + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 - 0.618T + T^{2} \) |
| 13 | \( 1 - 0.618iT - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - 1.61T + T^{2} \) |
| 23 | \( 1 + 1.61iT - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 0.618T + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.61iT - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.61iT - T^{2} \) |
| 83 | \( 1 - 2iT - T^{2} \) |
| 89 | \( 1 + 0.618T + T^{2} \) |
| 97 | \( 1 - 1.61iT - T^{2} \) |
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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
−9.342316748409619706469807320672, −8.558828613359730197367444142451, −7.66115581542961393213803750318, −6.73784434404710056966631771293, −6.21390436680707159320100339821, −5.18687773721832427217101701105, −4.08831533484336784572638961913, −3.13526401956997833404193791357, −2.41038003220776168509411392583, −1.15282602885632469257823728687,
1.47711377691477051913384929197, 2.84189087946226129861056641964, 3.47830670147833745669285728142, 5.00512660658087552940120361320, 5.65148346643809418653664292336, 6.26670983509683467230463290888, 7.24970083687714733744835500368, 7.78385315173531602656301683909, 8.550738355413035070585920306535, 9.403982628873421685553483037303