L(s) = 1 | − 3-s − 2·7-s + 9-s + 13-s − 6·17-s − 2·19-s + 2·21-s − 5·25-s − 27-s − 6·29-s − 2·31-s + 2·37-s − 39-s − 12·41-s + 4·43-s − 3·49-s + 6·51-s + 6·53-s + 2·57-s − 12·59-s + 2·61-s − 2·63-s + 10·67-s − 12·71-s + 14·73-s + 5·75-s − 8·79-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.277·13-s − 1.45·17-s − 0.458·19-s + 0.436·21-s − 25-s − 0.192·27-s − 1.11·29-s − 0.359·31-s + 0.328·37-s − 0.160·39-s − 1.87·41-s + 0.609·43-s − 3/7·49-s + 0.840·51-s + 0.824·53-s + 0.264·57-s − 1.56·59-s + 0.256·61-s − 0.251·63-s + 1.22·67-s − 1.42·71-s + 1.63·73-s + 0.577·75-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−10.21634971619086081222499193318, −9.385896632264966978634773161619, −8.531128945035369322451694692338, −7.32625330559643707687229250500, −6.49558938607564697804430447032, −5.75513068077913273540546960045, −4.55277073254635650297368891844, −3.54130855144350797063436114136, −2.01665432147547119953995139615, 0,
2.01665432147547119953995139615, 3.54130855144350797063436114136, 4.55277073254635650297368891844, 5.75513068077913273540546960045, 6.49558938607564697804430447032, 7.32625330559643707687229250500, 8.531128945035369322451694692338, 9.385896632264966978634773161619, 10.21634971619086081222499193318