L(s) = 1 | − 3-s − 5-s + 9-s + 2·11-s − 2·13-s + 15-s − 4·17-s − 8·23-s + 25-s − 27-s + 2·31-s − 2·33-s − 8·37-s + 2·39-s − 2·41-s − 2·43-s − 45-s − 10·47-s + 4·51-s + 2·53-s − 2·55-s + 4·59-s + 10·61-s + 2·65-s + 2·67-s + 8·69-s + 12·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.603·11-s − 0.554·13-s + 0.258·15-s − 0.970·17-s − 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.359·31-s − 0.348·33-s − 1.31·37-s + 0.320·39-s − 0.312·41-s − 0.304·43-s − 0.149·45-s − 1.45·47-s + 0.560·51-s + 0.274·53-s − 0.269·55-s + 0.520·59-s + 1.28·61-s + 0.248·65-s + 0.244·67-s + 0.963·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.84477661834915, −14.32315382044015, −13.90975622285297, −13.19149409522050, −12.77832513231054, −12.09634631984666, −11.75576451292348, −11.42779164702498, −10.71761157805326, −10.17125065441198, −9.769069024454439, −9.103351665359128, −8.452443428248007, −8.053738095378234, −7.316346387032013, −6.764411685728093, −6.382033966610232, −5.731349038911381, −4.909309203256513, −4.650912378801070, −3.749087690557155, −3.485911848388514, −2.254921040432007, −1.890096980876256, −0.7711789342877799, 0,
0.7711789342877799, 1.890096980876256, 2.254921040432007, 3.485911848388514, 3.749087690557155, 4.650912378801070, 4.909309203256513, 5.731349038911381, 6.382033966610232, 6.764411685728093, 7.316346387032013, 8.053738095378234, 8.452443428248007, 9.103351665359128, 9.769069024454439, 10.17125065441198, 10.71761157805326, 11.42779164702498, 11.75576451292348, 12.09634631984666, 12.77832513231054, 13.19149409522050, 13.90975622285297, 14.32315382044015, 14.84477661834915