L(s) = 1 | − 2.46·2-s − 2.74·3-s + 4.08·4-s + 0.488·5-s + 6.78·6-s − 1.50·7-s − 5.15·8-s + 4.55·9-s − 1.20·10-s + 11-s − 11.2·12-s − 1.24·13-s + 3.70·14-s − 1.34·15-s + 4.54·16-s + 3.29·17-s − 11.2·18-s + 4.48·19-s + 1.99·20-s + 4.12·21-s − 2.46·22-s + 2.86·23-s + 14.1·24-s − 4.76·25-s + 3.07·26-s − 4.27·27-s − 6.14·28-s + ⋯ |
L(s) = 1 | − 1.74·2-s − 1.58·3-s + 2.04·4-s + 0.218·5-s + 2.76·6-s − 0.567·7-s − 1.82·8-s + 1.51·9-s − 0.381·10-s + 0.301·11-s − 3.24·12-s − 0.345·13-s + 0.990·14-s − 0.346·15-s + 1.13·16-s + 0.799·17-s − 2.64·18-s + 1.02·19-s + 0.447·20-s + 0.900·21-s − 0.526·22-s + 0.597·23-s + 2.89·24-s − 0.952·25-s + 0.602·26-s − 0.822·27-s − 1.16·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3263814465\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3263814465\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - T \) |
| 131 | \( 1 + T \) |
good | 2 | \( 1 + 2.46T + 2T^{2} \) |
| 3 | \( 1 + 2.74T + 3T^{2} \) |
| 5 | \( 1 - 0.488T + 5T^{2} \) |
| 7 | \( 1 + 1.50T + 7T^{2} \) |
| 13 | \( 1 + 1.24T + 13T^{2} \) |
| 17 | \( 1 - 3.29T + 17T^{2} \) |
| 19 | \( 1 - 4.48T + 19T^{2} \) |
| 23 | \( 1 - 2.86T + 23T^{2} \) |
| 29 | \( 1 + 6.33T + 29T^{2} \) |
| 31 | \( 1 - 7.17T + 31T^{2} \) |
| 37 | \( 1 + 6.61T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 - 2.12T + 43T^{2} \) |
| 47 | \( 1 + 8.05T + 47T^{2} \) |
| 53 | \( 1 + 2.36T + 53T^{2} \) |
| 59 | \( 1 + 1.23T + 59T^{2} \) |
| 61 | \( 1 + 12.8T + 61T^{2} \) |
| 67 | \( 1 + 7.36T + 67T^{2} \) |
| 71 | \( 1 + 1.51T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 9.64T + 79T^{2} \) |
| 83 | \( 1 - 2.52T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 + 9.73T + 97T^{2} \) |
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
−9.679030788307486553796400083043, −9.036590875815633888418403457928, −7.74262649154241800328088575703, −7.28017873968845388880068905655, −6.31634172039404873373576290571, −5.84554891189254301252211489965, −4.75840340225696837605493742719, −3.16648265592704387863038212194, −1.63202910269640410666683401133, −0.59322275014551408390959607183,
0.59322275014551408390959607183, 1.63202910269640410666683401133, 3.16648265592704387863038212194, 4.75840340225696837605493742719, 5.84554891189254301252211489965, 6.31634172039404873373576290571, 7.28017873968845388880068905655, 7.74262649154241800328088575703, 9.036590875815633888418403457928, 9.679030788307486553796400083043