| L(s) = 1 | + 2·5-s − 4·11-s − 2·13-s − 17-s − 4·19-s − 4·23-s − 25-s + 2·29-s + 2·37-s − 10·41-s + 4·43-s + 8·47-s − 6·53-s − 8·55-s + 8·59-s − 2·61-s − 4·65-s + 4·67-s + 12·71-s + 10·73-s − 8·79-s − 2·85-s + 6·89-s − 8·95-s − 6·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.20·11-s − 0.554·13-s − 0.242·17-s − 0.917·19-s − 0.834·23-s − 1/5·25-s + 0.371·29-s + 0.328·37-s − 1.56·41-s + 0.609·43-s + 1.16·47-s − 0.824·53-s − 1.07·55-s + 1.04·59-s − 0.256·61-s − 0.496·65-s + 0.488·67-s + 1.42·71-s + 1.17·73-s − 0.900·79-s − 0.216·85-s + 0.635·89-s − 0.820·95-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.75402730452471, −13.39149871552510, −12.84655069361234, −12.44863857117192, −12.01885038346590, −11.24688771356191, −10.88417365473796, −10.27548312418580, −9.952518994636264, −9.611795829861672, −8.863714538722897, −8.405430128115408, −7.932261566467002, −7.390689887117682, −6.775734621118059, −6.256907400670067, −5.758215866263555, −5.261187161219608, −4.757673159571460, −4.163455451133952, −3.460125949253455, −2.715941349841548, −2.141906759778457, −1.921279916159363, −0.7775735373341982, 0,
0.7775735373341982, 1.921279916159363, 2.141906759778457, 2.715941349841548, 3.460125949253455, 4.163455451133952, 4.757673159571460, 5.261187161219608, 5.758215866263555, 6.256907400670067, 6.775734621118059, 7.390689887117682, 7.932261566467002, 8.405430128115408, 8.863714538722897, 9.611795829861672, 9.952518994636264, 10.27548312418580, 10.88417365473796, 11.24688771356191, 12.01885038346590, 12.44863857117192, 12.84655069361234, 13.39149871552510, 13.75402730452471
