L(s) = 1 | − 2-s + 4-s − 1.41·5-s − 1.41·7-s − 8-s + 1.41·10-s + 1.41·14-s + 16-s + 17-s − 1.41·20-s + 1.41·23-s + 1.00·25-s − 1.41·28-s + 1.41·29-s + 1.41·31-s − 32-s − 34-s + 2.00·35-s + 1.41·37-s + 1.41·40-s − 2·43-s − 1.41·46-s + 1.00·49-s − 1.00·50-s + 1.41·56-s − 1.41·58-s − 1.41·61-s + ⋯ |
L(s) = 1 | − 2-s + 4-s − 1.41·5-s − 1.41·7-s − 8-s + 1.41·10-s + 1.41·14-s + 16-s + 17-s − 1.41·20-s + 1.41·23-s + 1.00·25-s − 1.41·28-s + 1.41·29-s + 1.41·31-s − 32-s − 34-s + 2.00·35-s + 1.41·37-s + 1.41·40-s − 2·43-s − 1.41·46-s + 1.00·49-s − 1.00·50-s + 1.41·56-s − 1.41·58-s − 1.41·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4400220352\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4400220352\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + 1.41T + T^{2} \) |
| 7 | \( 1 + 1.41T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - 1.41T + T^{2} \) |
| 29 | \( 1 - 1.41T + T^{2} \) |
| 31 | \( 1 - 1.41T + T^{2} \) |
| 37 | \( 1 - 1.41T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 2T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + 1.41T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + 1.41T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.41T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - 2T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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
−9.921096026502799667245284448088, −9.089320330054712159635694496889, −8.262372192001780137580380421882, −7.61618971070998471370021541179, −6.78644340089740919807353642420, −6.16568914155503284830000562469, −4.71575497748483378375653844011, −3.36709884565606494745634365404, −2.93934477872671429160988624196, −0.835860832016285951941877200020,
0.835860832016285951941877200020, 2.93934477872671429160988624196, 3.36709884565606494745634365404, 4.71575497748483378375653844011, 6.16568914155503284830000562469, 6.78644340089740919807353642420, 7.61618971070998471370021541179, 8.262372192001780137580380421882, 9.089320330054712159635694496889, 9.921096026502799667245284448088