L(s) = 1 | − 3-s + 5-s + 4·7-s + 9-s − 4·11-s − 15-s + 6·17-s − 4·21-s + 4·23-s + 25-s − 27-s + 6·29-s − 8·31-s + 4·33-s + 4·35-s − 2·37-s − 10·41-s − 4·43-s + 45-s + 8·47-s + 9·49-s − 6·51-s + 2·53-s − 4·55-s − 4·59-s − 14·61-s + 4·63-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.258·15-s + 1.45·17-s − 0.872·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.696·33-s + 0.676·35-s − 0.328·37-s − 1.56·41-s − 0.609·43-s + 0.149·45-s + 1.16·47-s + 9/7·49-s − 0.840·51-s + 0.274·53-s − 0.539·55-s − 0.520·59-s − 1.79·61-s + 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.57939400761741, −12.93066728021694, −12.52073788071976, −12.04469534934629, −11.63434317852847, −11.03802262424380, −10.61700355641136, −10.38861064669170, −9.819707947317818, −9.202652195174414, −8.574791154793193, −8.193506244872693, −7.621717876725783, −7.333498570872979, −6.678288158121189, −6.006856573657259, −5.365807195238165, −5.178909081518382, −4.860230487891468, −4.089041166013696, −3.354810931688553, −2.779591458517302, −2.049266800247430, −1.472803089224481, −0.9718147750882420, 0,
0.9718147750882420, 1.472803089224481, 2.049266800247430, 2.779591458517302, 3.354810931688553, 4.089041166013696, 4.860230487891468, 5.178909081518382, 5.365807195238165, 6.006856573657259, 6.678288158121189, 7.333498570872979, 7.621717876725783, 8.193506244872693, 8.574791154793193, 9.202652195174414, 9.819707947317818, 10.38861064669170, 10.61700355641136, 11.03802262424380, 11.63434317852847, 12.04469534934629, 12.52073788071976, 12.93066728021694, 13.57939400761741