L(s) = 1 | − 3-s + 5-s − 7-s + 9-s − 6·13-s − 15-s + 2·17-s + 8·19-s + 21-s − 8·23-s + 25-s − 27-s − 2·29-s − 4·31-s − 35-s − 2·37-s + 6·39-s − 6·41-s − 4·43-s + 45-s − 8·47-s + 49-s − 2·51-s + 10·53-s − 8·57-s − 4·59-s − 2·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.66·13-s − 0.258·15-s + 0.485·17-s + 1.83·19-s + 0.218·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.169·35-s − 0.328·37-s + 0.960·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s − 1.16·47-s + 1/7·49-s − 0.280·51-s + 1.37·53-s − 1.05·57-s − 0.520·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−9.242967464128379637809586382976, −7.975429118345566696857391954617, −7.31447554122255167880309018857, −6.55815443374401606221787868995, −5.48396360051002103094444371014, −5.15942873071656623278244316377, −3.89765231085728538463298179402, −2.82363836469554907617267416973, −1.63265803663490016612214771155, 0,
1.63265803663490016612214771155, 2.82363836469554907617267416973, 3.89765231085728538463298179402, 5.15942873071656623278244316377, 5.48396360051002103094444371014, 6.55815443374401606221787868995, 7.31447554122255167880309018857, 7.975429118345566696857391954617, 9.242967464128379637809586382976