L(s) = 1 | + 0.618·2-s + 0.618·3-s − 0.618·4-s − 1.61·5-s + 0.381·6-s + 7-s − 8-s − 0.618·9-s − 1.00·10-s − 0.381·12-s + 0.618·14-s − 1.00·15-s + 17-s − 0.381·18-s + 0.999·20-s + 0.618·21-s − 0.618·24-s + 1.61·25-s − 27-s − 0.618·28-s − 0.618·30-s − 1.61·31-s + 0.999·32-s + 0.618·34-s − 1.61·35-s + 0.381·36-s + 1.61·40-s + ⋯ |
L(s) = 1 | + 0.618·2-s + 0.618·3-s − 0.618·4-s − 1.61·5-s + 0.381·6-s + 7-s − 8-s − 0.618·9-s − 1.00·10-s − 0.381·12-s + 0.618·14-s − 1.00·15-s + 17-s − 0.381·18-s + 0.999·20-s + 0.618·21-s − 0.618·24-s + 1.61·25-s − 27-s − 0.618·28-s − 0.618·30-s − 1.61·31-s + 0.999·32-s + 0.618·34-s − 1.61·35-s + 0.381·36-s + 1.61·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6840384900\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6840384900\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 2 | \( 1 - 0.618T + T^{2} \) |
| 3 | \( 1 - 0.618T + T^{2} \) |
| 5 | \( 1 + 1.61T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.61T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 0.618T + T^{2} \) |
| 43 | \( 1 - 0.618T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.61T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 0.618T + T^{2} \) |
| 67 | \( 1 + 1.61T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 0.618T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + 1.61T + T^{2} \) |
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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
−14.04573251757204190644351240073, −12.68729914443010867183501235312, −11.84702716241586137927181515730, −10.99785240360884651423191253530, −9.197151922943741932582102142604, −8.252672509741494813962148682655, −7.59622314204297360302731343397, −5.48989736619258142965789236319, −4.26497223589162521783933658687, −3.27448017805867294444555410081,
3.27448017805867294444555410081, 4.26497223589162521783933658687, 5.48989736619258142965789236319, 7.59622314204297360302731343397, 8.252672509741494813962148682655, 9.197151922943741932582102142604, 10.99785240360884651423191253530, 11.84702716241586137927181515730, 12.68729914443010867183501235312, 14.04573251757204190644351240073