L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s + 11-s − 12-s − 3.46·13-s + 14-s + 15-s + 16-s + 3.46·17-s + 18-s − 20-s − 21-s + 22-s + 5.46·23-s − 24-s + 25-s − 3.46·26-s − 27-s + 28-s + 2·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s − 0.960·13-s + 0.267·14-s + 0.258·15-s + 0.250·16-s + 0.840·17-s + 0.235·18-s − 0.223·20-s − 0.218·21-s + 0.213·22-s + 1.13·23-s − 0.204·24-s + 0.200·25-s − 0.679·26-s − 0.192·27-s + 0.188·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.258446001\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.258446001\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 5.46T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 - 0.535T + 37T^{2} \) |
| 41 | \( 1 - 4.92T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + 0.928T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 - 2.53T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 6.92T + 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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
−9.055268337941800597375419038106, −7.963035052772262009500208719236, −7.32205501542056121460323582833, −6.69120368070970301283269534443, −5.63006098372896440713622362773, −5.07976434549722291831034269642, −4.27744913859288882775878299989, −3.39665751868637659102151119750, −2.28365214895090924182892509736, −0.924254213124414443910935208215,
0.924254213124414443910935208215, 2.28365214895090924182892509736, 3.39665751868637659102151119750, 4.27744913859288882775878299989, 5.07976434549722291831034269642, 5.63006098372896440713622362773, 6.69120368070970301283269534443, 7.32205501542056121460323582833, 7.963035052772262009500208719236, 9.055268337941800597375419038106