L(s) = 1 | + 2·13-s + 6·17-s + 8·19-s − 6·29-s − 4·31-s + 10·37-s − 6·41-s − 4·43-s − 6·53-s + 12·59-s + 10·61-s − 4·67-s + 12·71-s − 10·73-s − 8·79-s + 12·83-s − 6·89-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | + 0.554·13-s + 1.45·17-s + 1.83·19-s − 1.11·29-s − 0.718·31-s + 1.64·37-s − 0.937·41-s − 0.609·43-s − 0.824·53-s + 1.56·59-s + 1.28·61-s − 0.488·67-s + 1.42·71-s − 1.17·73-s − 0.900·79-s + 1.31·83-s − 0.635·89-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.156920464\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.156920464\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 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 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.30912603691762, −12.68647625537679, −12.26060063528297, −11.65836689792462, −11.35650661599666, −10.99639118652184, −10.12882293079986, −9.884167646086987, −9.498184703293588, −8.935614631813923, −8.299716026603124, −7.881358410738993, −7.406840957496248, −7.018338932395977, −6.298777926797488, −5.737854248383668, −5.348252439510580, −4.981206914612833, −4.056353987023267, −3.621639900926514, −3.188395207066793, −2.575525016394423, −1.700758950809128, −1.201406989531250, −0.5556913153217068,
0.5556913153217068, 1.201406989531250, 1.700758950809128, 2.575525016394423, 3.188395207066793, 3.621639900926514, 4.056353987023267, 4.981206914612833, 5.348252439510580, 5.737854248383668, 6.298777926797488, 7.018338932395977, 7.406840957496248, 7.881358410738993, 8.299716026603124, 8.935614631813923, 9.498184703293588, 9.884167646086987, 10.12882293079986, 10.99639118652184, 11.35650661599666, 11.65836689792462, 12.26060063528297, 12.68647625537679, 13.30912603691762