| L(s) = 1 | + 7-s + 6·11-s + 5·13-s + 6·17-s − 5·19-s − 6·23-s + 6·29-s + 31-s + 2·37-s + 43-s + 6·47-s − 6·49-s + 12·53-s − 6·59-s − 13·61-s − 11·67-s + 2·73-s + 6·77-s − 8·79-s − 6·83-s + 5·91-s − 7·97-s + 12·101-s + 4·103-s + 12·107-s − 7·109-s − 12·113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.80·11-s + 1.38·13-s + 1.45·17-s − 1.14·19-s − 1.25·23-s + 1.11·29-s + 0.179·31-s + 0.328·37-s + 0.152·43-s + 0.875·47-s − 6/7·49-s + 1.64·53-s − 0.781·59-s − 1.66·61-s − 1.34·67-s + 0.234·73-s + 0.683·77-s − 0.900·79-s − 0.658·83-s + 0.524·91-s − 0.710·97-s + 1.19·101-s + 0.394·103-s + 1.16·107-s − 0.670·109-s − 1.12·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.506340872\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.506340872\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−8.557863511057997757970268824560, −7.961542507896363173235765905676, −7.00797354199330012097468426598, −6.11097472346481350589046305204, −5.91097970406424547476593156462, −4.48877495654256184592709120945, −3.99298778613570722908140056723, −3.15734679160146745553465738936, −1.76993182691893440967496910692, −1.03428119739066030819519430786,
1.03428119739066030819519430786, 1.76993182691893440967496910692, 3.15734679160146745553465738936, 3.99298778613570722908140056723, 4.48877495654256184592709120945, 5.91097970406424547476593156462, 6.11097472346481350589046305204, 7.00797354199330012097468426598, 7.961542507896363173235765905676, 8.557863511057997757970268824560
