L(s) = 1 | − 1.61·2-s + 0.618·4-s + 5-s − 1.23·7-s + 2.23·8-s − 1.61·10-s + 0.763·13-s + 2.00·14-s − 4.85·16-s − 3.47·17-s − 2.47·19-s + 0.618·20-s − 8.70·23-s + 25-s − 1.23·26-s − 0.763·28-s − 3.23·29-s + 6.70·31-s + 3.38·32-s + 5.61·34-s − 1.23·35-s + 5.23·37-s + 4.00·38-s + 2.23·40-s − 12.4·41-s − 0.763·43-s + 14.0·46-s + ⋯ |
L(s) = 1 | − 1.14·2-s + 0.309·4-s + 0.447·5-s − 0.467·7-s + 0.790·8-s − 0.511·10-s + 0.211·13-s + 0.534·14-s − 1.21·16-s − 0.842·17-s − 0.567·19-s + 0.138·20-s − 1.81·23-s + 0.200·25-s − 0.242·26-s − 0.144·28-s − 0.600·29-s + 1.20·31-s + 0.597·32-s + 0.963·34-s − 0.208·35-s + 0.860·37-s + 0.648·38-s + 0.353·40-s − 1.94·41-s − 0.116·43-s + 2.07·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6864585471\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6864585471\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 13 | \( 1 - 0.763T + 13T^{2} \) |
| 17 | \( 1 + 3.47T + 17T^{2} \) |
| 19 | \( 1 + 2.47T + 19T^{2} \) |
| 23 | \( 1 + 8.70T + 23T^{2} \) |
| 29 | \( 1 + 3.23T + 29T^{2} \) |
| 31 | \( 1 - 6.70T + 31T^{2} \) |
| 37 | \( 1 - 5.23T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 + 0.763T + 43T^{2} \) |
| 47 | \( 1 - 4.70T + 47T^{2} \) |
| 53 | \( 1 - 7.94T + 53T^{2} \) |
| 59 | \( 1 - 1.70T + 59T^{2} \) |
| 61 | \( 1 - 7.47T + 61T^{2} \) |
| 67 | \( 1 - 6.76T + 67T^{2} \) |
| 71 | \( 1 - 5.52T + 71T^{2} \) |
| 73 | \( 1 + 1.52T + 73T^{2} \) |
| 79 | \( 1 + 0.708T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 9.23T + 89T^{2} \) |
| 97 | \( 1 - 17.2T + 97T^{2} \) |
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
−8.319160630156221569246901543711, −7.69853940835635984977442426168, −6.71378759289885202733974485972, −6.34194502106037896693749929805, −5.35607978700019426789431952399, −4.43101098685752483084996813196, −3.73256709960995503474593628985, −2.42972087965228013820272989530, −1.77379086665358460514946281786, −0.52362554190357069146652618494,
0.52362554190357069146652618494, 1.77379086665358460514946281786, 2.42972087965228013820272989530, 3.73256709960995503474593628985, 4.43101098685752483084996813196, 5.35607978700019426789431952399, 6.34194502106037896693749929805, 6.71378759289885202733974485972, 7.69853940835635984977442426168, 8.319160630156221569246901543711