L(s) = 1 | + 0.618·2-s − 1.61·3-s − 0.618·4-s − 1.00·6-s + 0.618·7-s − 8-s + 1.61·9-s + 0.999·12-s + 0.381·14-s − 1.61·17-s + 1.00·18-s − 1.00·21-s + 1.61·24-s + 25-s − 27-s − 0.381·28-s + 0.999·32-s − 1.00·34-s − 0.999·36-s − 1.61·37-s − 0.618·42-s + 47-s − 0.618·49-s + 0.618·50-s + 2.61·51-s + 0.618·53-s − 0.618·54-s + ⋯ |
L(s) = 1 | + 0.618·2-s − 1.61·3-s − 0.618·4-s − 1.00·6-s + 0.618·7-s − 8-s + 1.61·9-s + 0.999·12-s + 0.381·14-s − 1.61·17-s + 1.00·18-s − 1.00·21-s + 1.61·24-s + 25-s − 27-s − 0.381·28-s + 0.999·32-s − 1.00·34-s − 0.999·36-s − 1.61·37-s − 0.618·42-s + 47-s − 0.618·49-s + 0.618·50-s + 2.61·51-s + 0.618·53-s − 0.618·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3597283851\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3597283851\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 47 | \( 1 - T \) |
good | 2 | \( 1 - 0.618T + T^{2} \) |
| 3 | \( 1 + 1.61T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - 0.618T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.61T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.61T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 53 | \( 1 - 0.618T + T^{2} \) |
| 59 | \( 1 - 0.618T + T^{2} \) |
| 61 | \( 1 - 0.618T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 1.61T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 1.61T + T^{2} \) |
| 83 | \( 1 - 2T + T^{2} \) |
| 89 | \( 1 - 0.618T + T^{2} \) |
| 97 | \( 1 - 0.618T + T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.07886948847836249116832192876, −14.90531002592942505063566570766, −13.54757720598963822818329142040, −12.50277346124156882218469552755, −11.52817428198532142050518472672, −10.50369075022612179601131080455, −8.810223959306080153905100573024, −6.73380655474091416962152422615, −5.40411667131609441039217306830, −4.44331727518334815292959324093,
4.44331727518334815292959324093, 5.40411667131609441039217306830, 6.73380655474091416962152422615, 8.810223959306080153905100573024, 10.50369075022612179601131080455, 11.52817428198532142050518472672, 12.50277346124156882218469552755, 13.54757720598963822818329142040, 14.90531002592942505063566570766, 16.07886948847836249116832192876