L(s) = 1 | − 7-s − 2·13-s − 8·17-s + 6·19-s − 8·23-s − 5·25-s − 6·29-s + 10·31-s + 2·37-s − 8·41-s + 4·43-s − 2·47-s + 49-s − 2·53-s + 8·59-s − 2·61-s + 4·67-s + 4·71-s + 4·73-s + 8·79-s − 18·83-s − 2·89-s + 2·91-s − 14·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 0.554·13-s − 1.94·17-s + 1.37·19-s − 1.66·23-s − 25-s − 1.11·29-s + 1.79·31-s + 0.328·37-s − 1.24·41-s + 0.609·43-s − 0.291·47-s + 1/7·49-s − 0.274·53-s + 1.04·59-s − 0.256·61-s + 0.488·67-s + 0.474·71-s + 0.468·73-s + 0.900·79-s − 1.97·83-s − 0.211·89-s + 0.209·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7228321163\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7228321163\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.01311546635561, −13.82843002055587, −13.41929070756521, −12.80933310237664, −12.19836339181941, −11.75662862142621, −11.35154826345109, −10.80033159569254, −9.953642278430029, −9.774419359571346, −9.341891150937885, −8.502086889072131, −8.147720152092535, −7.523300821676577, −6.913540571960356, −6.474759174039291, −5.840331431273919, −5.315954237034663, −4.572991645529142, −4.076434939532783, −3.505406111280672, −2.608263672964768, −2.215099766165204, −1.394940176292454, −0.2819100172794180,
0.2819100172794180, 1.394940176292454, 2.215099766165204, 2.608263672964768, 3.505406111280672, 4.076434939532783, 4.572991645529142, 5.315954237034663, 5.840331431273919, 6.474759174039291, 6.913540571960356, 7.523300821676577, 8.147720152092535, 8.502086889072131, 9.341891150937885, 9.774419359571346, 9.953642278430029, 10.80033159569254, 11.35154826345109, 11.75662862142621, 12.19836339181941, 12.80933310237664, 13.41929070756521, 13.82843002055587, 14.01311546635561