L(s) = 1 | − 3·5-s + 7-s − 6·11-s − 4·13-s − 19-s + 4·23-s + 4·25-s + 6·29-s − 3·35-s + 12·37-s − 6·41-s − 4·43-s − 9·47-s − 6·49-s + 5·53-s + 18·55-s − 61-s + 12·65-s − 10·67-s − 12·71-s + 11·73-s − 6·77-s − 12·79-s − 4·83-s − 14·89-s − 4·91-s + 3·95-s + ⋯ |
L(s) = 1 | − 1.34·5-s + 0.377·7-s − 1.80·11-s − 1.10·13-s − 0.229·19-s + 0.834·23-s + 4/5·25-s + 1.11·29-s − 0.507·35-s + 1.97·37-s − 0.937·41-s − 0.609·43-s − 1.31·47-s − 6/7·49-s + 0.686·53-s + 2.42·55-s − 0.128·61-s + 1.48·65-s − 1.22·67-s − 1.42·71-s + 1.28·73-s − 0.683·77-s − 1.35·79-s − 0.439·83-s − 1.48·89-s − 0.419·91-s + 0.307·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 463 | \( 1 + T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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
−14.06190031360102, −13.35304244467279, −12.89298449369798, −12.67290417287411, −11.92443967983242, −11.53795724930566, −11.22135367714988, −10.54710208104922, −10.10395260226906, −9.748866049090142, −8.821324517157341, −8.405448999661298, −7.925813364854063, −7.586125398245377, −7.168193452776972, −6.515317426190602, −5.776823466512263, −5.072106860887832, −4.683766353034323, −4.423875543696565, −3.446430094750294, −2.906091933652088, −2.536784905006263, −1.622567650048351, −0.6095319619899024, 0,
0.6095319619899024, 1.622567650048351, 2.536784905006263, 2.906091933652088, 3.446430094750294, 4.423875543696565, 4.683766353034323, 5.072106860887832, 5.776823466512263, 6.515317426190602, 7.168193452776972, 7.586125398245377, 7.925813364854063, 8.405448999661298, 8.821324517157341, 9.748866049090142, 10.10395260226906, 10.54710208104922, 11.22135367714988, 11.53795724930566, 11.92443967983242, 12.67290417287411, 12.89298449369798, 13.35304244467279, 14.06190031360102