L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 4·7-s − 8-s + 9-s − 12-s + 13-s − 4·14-s + 16-s + 17-s − 18-s − 2·19-s − 4·21-s + 4·23-s + 24-s − 5·25-s − 26-s − 27-s + 4·28-s + 2·29-s − 32-s − 34-s + 36-s + 10·37-s + 2·38-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 0.277·13-s − 1.06·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.458·19-s − 0.872·21-s + 0.834·23-s + 0.204·24-s − 25-s − 0.196·26-s − 0.192·27-s + 0.755·28-s + 0.371·29-s − 0.176·32-s − 0.171·34-s + 1/6·36-s + 1.64·37-s + 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.768014538\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.768014538\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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
−13.31139971863781, −12.55927709555341, −12.24089718472371, −11.72422063703304, −11.18301586723950, −10.93358035423300, −10.66535842440492, −9.863883433456573, −9.520532813807682, −8.910376950470601, −8.381093602459590, −8.019170549744929, −7.416954844681398, −7.217045241790040, −6.319757927763505, −5.950611277969644, −5.460509439135345, −4.827880580609468, −4.302122903958263, −3.905872278277434, −2.837476664317954, −2.432376911221302, −1.531523834351364, −1.275919053910483, −0.4829107896071797,
0.4829107896071797, 1.275919053910483, 1.531523834351364, 2.432376911221302, 2.837476664317954, 3.905872278277434, 4.302122903958263, 4.827880580609468, 5.460509439135345, 5.950611277969644, 6.319757927763505, 7.217045241790040, 7.416954844681398, 8.019170549744929, 8.381093602459590, 8.910376950470601, 9.520532813807682, 9.863883433456573, 10.66535842440492, 10.93358035423300, 11.18301586723950, 11.72422063703304, 12.24089718472371, 12.55927709555341, 13.31139971863781