L(s) = 1 | − 5-s + 7-s − 3·9-s + 2·11-s + 13-s − 4·17-s − 4·19-s + 4·23-s + 25-s + 2·29-s − 2·31-s − 35-s − 6·37-s + 2·43-s + 3·45-s − 8·47-s + 49-s + 12·53-s − 2·55-s + 4·59-s + 2·61-s − 3·63-s − 65-s + 16·67-s − 12·71-s + 6·73-s + 2·77-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.377·7-s − 9-s + 0.603·11-s + 0.277·13-s − 0.970·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s + 0.371·29-s − 0.359·31-s − 0.169·35-s − 0.986·37-s + 0.304·43-s + 0.447·45-s − 1.16·47-s + 1/7·49-s + 1.64·53-s − 0.269·55-s + 0.520·59-s + 0.256·61-s − 0.377·63-s − 0.124·65-s + 1.95·67-s − 1.42·71-s + 0.702·73-s + 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−15.34274676062562, −14.78910445604298, −14.58033834556426, −13.85126566150195, −13.37050506642358, −12.77952727141265, −12.17460808928961, −11.56815751579892, −11.25664287019067, −10.73637383109033, −10.18888883580771, −9.230466198405256, −8.866974576139000, −8.407518364062865, −7.935957799369266, −6.937855970634521, −6.741662493001865, −5.967887431896378, −5.279577881183802, −4.710006184775839, −3.979359995720126, −3.460061258337159, −2.589172881891947, −1.969313638728767, −0.9478674089519174, 0,
0.9478674089519174, 1.969313638728767, 2.589172881891947, 3.460061258337159, 3.979359995720126, 4.710006184775839, 5.279577881183802, 5.967887431896378, 6.741662493001865, 6.937855970634521, 7.935957799369266, 8.407518364062865, 8.866974576139000, 9.230466198405256, 10.18888883580771, 10.73637383109033, 11.25664287019067, 11.56815751579892, 12.17460808928961, 12.77952727141265, 13.37050506642358, 13.85126566150195, 14.58033834556426, 14.78910445604298, 15.34274676062562