| L(s) = 1 | + 0.324·2-s + 2.83·3-s − 1.89·4-s + 2.69·5-s + 0.919·6-s + 3.71·7-s − 1.26·8-s + 5.02·9-s + 0.873·10-s + 0.340·11-s − 5.36·12-s − 4.32·13-s + 1.20·14-s + 7.63·15-s + 3.37·16-s − 17-s + 1.63·18-s + 1.00·19-s − 5.10·20-s + 10.5·21-s + 0.110·22-s − 5.84·23-s − 3.58·24-s + 2.25·25-s − 1.40·26-s + 5.75·27-s − 7.03·28-s + ⋯ |
| L(s) = 1 | + 0.229·2-s + 1.63·3-s − 0.947·4-s + 1.20·5-s + 0.375·6-s + 1.40·7-s − 0.446·8-s + 1.67·9-s + 0.276·10-s + 0.102·11-s − 1.54·12-s − 1.19·13-s + 0.322·14-s + 1.97·15-s + 0.844·16-s − 0.242·17-s + 0.384·18-s + 0.231·19-s − 1.14·20-s + 2.29·21-s + 0.0235·22-s − 1.21·23-s − 0.730·24-s + 0.450·25-s − 0.275·26-s + 1.10·27-s − 1.33·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.334303269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.334303269\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 2 | \( 1 - 0.324T + 2T^{2} \) |
| 3 | \( 1 - 2.83T + 3T^{2} \) |
| 5 | \( 1 - 2.69T + 5T^{2} \) |
| 7 | \( 1 - 3.71T + 7T^{2} \) |
| 11 | \( 1 - 0.340T + 11T^{2} \) |
| 13 | \( 1 + 4.32T + 13T^{2} \) |
| 19 | \( 1 - 1.00T + 19T^{2} \) |
| 23 | \( 1 + 5.84T + 23T^{2} \) |
| 29 | \( 1 - 7.99T + 29T^{2} \) |
| 31 | \( 1 + 0.280T + 31T^{2} \) |
| 37 | \( 1 + 4.98T + 37T^{2} \) |
| 41 | \( 1 + 9.93T + 41T^{2} \) |
| 43 | \( 1 + 5.47T + 43T^{2} \) |
| 47 | \( 1 + 7.98T + 47T^{2} \) |
| 53 | \( 1 - 8.66T + 53T^{2} \) |
| 61 | \( 1 + 8.26T + 61T^{2} \) |
| 67 | \( 1 - 15.0T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 - 6.40T + 73T^{2} \) |
| 79 | \( 1 + 1.31T + 79T^{2} \) |
| 83 | \( 1 + 6.78T + 83T^{2} \) |
| 89 | \( 1 - 4.00T + 89T^{2} \) |
| 97 | \( 1 - 17.0T + 97T^{2} \) |
| 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
−9.910179865260252381789961124463, −9.070505799679112946828003112618, −8.332462475259644986506619564026, −7.914412067628878809200735514986, −6.68021259278003577946325789385, −5.26915761360903390999383835640, −4.74212900950635988626275770588, −3.65128391028622607450551247377, −2.43468095638106260019193777494, −1.66215043763535746900439774554,
1.66215043763535746900439774554, 2.43468095638106260019193777494, 3.65128391028622607450551247377, 4.74212900950635988626275770588, 5.26915761360903390999383835640, 6.68021259278003577946325789385, 7.914412067628878809200735514986, 8.332462475259644986506619564026, 9.070505799679112946828003112618, 9.910179865260252381789961124463
