| L(s) = 1 | − 5-s − 2·11-s + 2·13-s + 4·19-s + 2·23-s + 25-s − 6·29-s + 2·37-s + 12·41-s + 43-s + 2·47-s − 7·49-s + 8·53-s + 2·55-s − 2·59-s − 2·61-s − 2·65-s − 4·67-s + 8·71-s − 14·73-s − 8·79-s + 14·83-s + 6·89-s − 4·95-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.603·11-s + 0.554·13-s + 0.917·19-s + 0.417·23-s + 1/5·25-s − 1.11·29-s + 0.328·37-s + 1.87·41-s + 0.152·43-s + 0.291·47-s − 49-s + 1.09·53-s + 0.269·55-s − 0.260·59-s − 0.256·61-s − 0.248·65-s − 0.488·67-s + 0.949·71-s − 1.63·73-s − 0.900·79-s + 1.53·83-s + 0.635·89-s − 0.410·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.882356608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.882356608\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.02752410939501, −14.66063106902981, −14.08637580558232, −13.33571424953636, −13.11382694795070, −12.48453089962052, −11.82946994857990, −11.38672282226420, −10.83213018011022, −10.41947372757244, −9.601383608609765, −9.182140535946283, −8.609435660382062, −7.796368710302901, −7.596034269679480, −6.940955549800568, −6.119169283097090, −5.642195510359317, −4.995944148694824, −4.306167766585943, −3.661936918637817, −3.028053494125666, −2.334725034523500, −1.364540075478151, −0.5536523934889113,
0.5536523934889113, 1.364540075478151, 2.334725034523500, 3.028053494125666, 3.661936918637817, 4.306167766585943, 4.995944148694824, 5.642195510359317, 6.119169283097090, 6.940955549800568, 7.596034269679480, 7.796368710302901, 8.609435660382062, 9.182140535946283, 9.601383608609765, 10.41947372757244, 10.83213018011022, 11.38672282226420, 11.82946994857990, 12.48453089962052, 13.11382694795070, 13.33571424953636, 14.08637580558232, 14.66063106902981, 15.02752410939501
