| L(s) = 1 | − 3-s − 2.63·5-s + 0.604·7-s + 9-s − 4.18·11-s − 0.588·13-s + 2.63·15-s + 3.13·17-s − 3.33·19-s − 0.604·21-s − 2.06·23-s + 1.96·25-s − 27-s + 10.1·29-s + 1.00·31-s + 4.18·33-s − 1.59·35-s − 0.963·37-s + 0.588·39-s + 10.0·41-s + 1.46·43-s − 2.63·45-s + 3.81·47-s − 6.63·49-s − 3.13·51-s − 2.52·53-s + 11.0·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.18·5-s + 0.228·7-s + 0.333·9-s − 1.26·11-s − 0.163·13-s + 0.681·15-s + 0.760·17-s − 0.764·19-s − 0.131·21-s − 0.429·23-s + 0.392·25-s − 0.192·27-s + 1.88·29-s + 0.181·31-s + 0.728·33-s − 0.269·35-s − 0.158·37-s + 0.0941·39-s + 1.57·41-s + 0.223·43-s − 0.393·45-s + 0.556·47-s − 0.947·49-s − 0.439·51-s − 0.347·53-s + 1.48·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 2.63T + 5T^{2} \) |
| 7 | \( 1 - 0.604T + 7T^{2} \) |
| 11 | \( 1 + 4.18T + 11T^{2} \) |
| 13 | \( 1 + 0.588T + 13T^{2} \) |
| 17 | \( 1 - 3.13T + 17T^{2} \) |
| 19 | \( 1 + 3.33T + 19T^{2} \) |
| 23 | \( 1 + 2.06T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 - 1.00T + 31T^{2} \) |
| 37 | \( 1 + 0.963T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 1.46T + 43T^{2} \) |
| 47 | \( 1 - 3.81T + 47T^{2} \) |
| 53 | \( 1 + 2.52T + 53T^{2} \) |
| 59 | \( 1 - 3.70T + 59T^{2} \) |
| 61 | \( 1 - 0.962T + 61T^{2} \) |
| 67 | \( 1 - 6.12T + 67T^{2} \) |
| 71 | \( 1 + 4.34T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 + 9.16T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 - 5.57T + 89T^{2} \) |
| 97 | \( 1 + 6.60T + 97T^{2} \) |
| 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
−7.59170847268746726032001547624, −6.88016383263570607197281259292, −6.06293741428409776471106721221, −5.34643820257339905080062936407, −4.60985436769278094863459942587, −4.10006420506102741813520079466, −3.12095647172877420212293083938, −2.32530607958337299293388299067, −0.964502285674061218576026429436, 0,
0.964502285674061218576026429436, 2.32530607958337299293388299067, 3.12095647172877420212293083938, 4.10006420506102741813520079466, 4.60985436769278094863459942587, 5.34643820257339905080062936407, 6.06293741428409776471106721221, 6.88016383263570607197281259292, 7.59170847268746726032001547624
