L(s) = 1 | − 2.41·3-s + 1.82·5-s − 4·7-s + 2.82·9-s + 0.414·11-s − 6.65·13-s − 4.41·15-s − 7.65·17-s + 2·19-s + 9.65·21-s + 0.828·23-s − 1.65·25-s + 0.414·27-s + 29-s − 5.58·31-s − 0.999·33-s − 7.31·35-s + 9.65·37-s + 16.0·39-s + 1.65·41-s + 4.41·43-s + 5.17·45-s + 2.07·47-s + 9·49-s + 18.4·51-s − 7·53-s + 0.757·55-s + ⋯ |
L(s) = 1 | − 1.39·3-s + 0.817·5-s − 1.51·7-s + 0.942·9-s + 0.124·11-s − 1.84·13-s − 1.13·15-s − 1.85·17-s + 0.458·19-s + 2.10·21-s + 0.172·23-s − 0.331·25-s + 0.0797·27-s + 0.185·29-s − 1.00·31-s − 0.174·33-s − 1.23·35-s + 1.58·37-s + 2.57·39-s + 0.258·41-s + 0.673·43-s + 0.770·45-s + 0.302·47-s + 1.28·49-s + 2.58·51-s − 0.961·53-s + 0.102·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 - T \) |
good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 5 | \( 1 - 1.82T + 5T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 - 0.414T + 11T^{2} \) |
| 13 | \( 1 + 6.65T + 13T^{2} \) |
| 17 | \( 1 + 7.65T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 0.828T + 23T^{2} \) |
| 31 | \( 1 + 5.58T + 31T^{2} \) |
| 37 | \( 1 - 9.65T + 37T^{2} \) |
| 41 | \( 1 - 1.65T + 41T^{2} \) |
| 43 | \( 1 - 4.41T + 43T^{2} \) |
| 47 | \( 1 - 2.07T + 47T^{2} \) |
| 53 | \( 1 + 7T + 53T^{2} \) |
| 59 | \( 1 + 6.48T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 + 3.17T + 83T^{2} \) |
| 89 | \( 1 - 1.65T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
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
−11.73338682791196080835853331094, −10.75604978898843057938215184378, −9.776283104086985290268219342624, −9.281831796137677005691996565079, −7.22449183949528396811369043833, −6.44152563531466937773900159305, −5.66240382862090237604937633731, −4.51392614233037072420827087027, −2.54992374104922361557245568809, 0,
2.54992374104922361557245568809, 4.51392614233037072420827087027, 5.66240382862090237604937633731, 6.44152563531466937773900159305, 7.22449183949528396811369043833, 9.281831796137677005691996565079, 9.776283104086985290268219342624, 10.75604978898843057938215184378, 11.73338682791196080835853331094