| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s − 11-s + 2·13-s + 15-s + 8·17-s − 2·19-s + 2·21-s + 8·23-s + 25-s − 27-s + 33-s + 2·35-s + 2·37-s − 2·39-s + 6·43-s − 45-s + 8·47-s − 3·49-s − 8·51-s + 6·53-s + 55-s + 2·57-s − 4·59-s + 10·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.258·15-s + 1.94·17-s − 0.458·19-s + 0.436·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.174·33-s + 0.338·35-s + 0.328·37-s − 0.320·39-s + 0.914·43-s − 0.149·45-s + 1.16·47-s − 3/7·49-s − 1.12·51-s + 0.824·53-s + 0.134·55-s + 0.264·57-s − 0.520·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.070023212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.070023212\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−10.58917076131701257357177911550, −9.788620372840369960521330671071, −8.850361822859748464532686119818, −7.78412586857067234094953487514, −6.98759403795642026924393450017, −5.99158381383684602539992031128, −5.17810728666522882356486400345, −3.91288399319096089362633398104, −2.94270272449041975847268775885, −0.938183538879467721656887858535,
0.938183538879467721656887858535, 2.94270272449041975847268775885, 3.91288399319096089362633398104, 5.17810728666522882356486400345, 5.99158381383684602539992031128, 6.98759403795642026924393450017, 7.78412586857067234094953487514, 8.850361822859748464532686119818, 9.788620372840369960521330671071, 10.58917076131701257357177911550
