L(s) = 1 | + 0.169·2-s − 1.63·3-s − 1.97·4-s − 3.37·5-s − 0.277·6-s + 0.890·7-s − 0.675·8-s − 0.335·9-s − 0.574·10-s − 2.91·11-s + 3.21·12-s − 4.89·13-s + 0.151·14-s + 5.51·15-s + 3.82·16-s − 17-s − 0.0571·18-s + 7.26·19-s + 6.65·20-s − 1.45·21-s − 0.494·22-s − 8.96·23-s + 1.10·24-s + 6.40·25-s − 0.832·26-s + 5.44·27-s − 1.75·28-s + ⋯ |
L(s) = 1 | + 0.120·2-s − 0.942·3-s − 0.985·4-s − 1.51·5-s − 0.113·6-s + 0.336·7-s − 0.238·8-s − 0.111·9-s − 0.181·10-s − 0.877·11-s + 0.928·12-s − 1.35·13-s + 0.0404·14-s + 1.42·15-s + 0.956·16-s − 0.242·17-s − 0.0134·18-s + 1.66·19-s + 1.48·20-s − 0.317·21-s − 0.105·22-s − 1.86·23-s + 0.224·24-s + 1.28·25-s − 0.163·26-s + 1.04·27-s − 0.331·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3203618578\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3203618578\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + T \) |
| 43 | \( 1 - T \) |
good | 2 | \( 1 - 0.169T + 2T^{2} \) |
| 3 | \( 1 + 1.63T + 3T^{2} \) |
| 5 | \( 1 + 3.37T + 5T^{2} \) |
| 7 | \( 1 - 0.890T + 7T^{2} \) |
| 11 | \( 1 + 2.91T + 11T^{2} \) |
| 13 | \( 1 + 4.89T + 13T^{2} \) |
| 19 | \( 1 - 7.26T + 19T^{2} \) |
| 23 | \( 1 + 8.96T + 23T^{2} \) |
| 29 | \( 1 - 5.06T + 29T^{2} \) |
| 31 | \( 1 - 4.35T + 31T^{2} \) |
| 37 | \( 1 + 0.601T + 37T^{2} \) |
| 41 | \( 1 + 5.08T + 41T^{2} \) |
| 47 | \( 1 + 3.67T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 3.54T + 61T^{2} \) |
| 67 | \( 1 - 3.39T + 67T^{2} \) |
| 71 | \( 1 + 6.22T + 71T^{2} \) |
| 73 | \( 1 + 0.844T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 + 9.55T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.36341842014000570491908562492, −9.751700033885912902348383819094, −8.332233567971384699184518023782, −7.978266639597404486437576847095, −7.00233433006422776440282023583, −5.57068214630569752681675472848, −4.95205663885520368007652791198, −4.19605520192016096243327443772, −2.97127220302087397467172509328, −0.46803698963875958447629706692,
0.46803698963875958447629706692, 2.97127220302087397467172509328, 4.19605520192016096243327443772, 4.95205663885520368007652791198, 5.57068214630569752681675472848, 7.00233433006422776440282023583, 7.978266639597404486437576847095, 8.332233567971384699184518023782, 9.751700033885912902348383819094, 10.36341842014000570491908562492