| L(s) = 1 | + 5-s + 7-s + 6·13-s + 2·17-s − 4·19-s + 4·23-s + 25-s − 6·29-s + 35-s + 6·37-s + 2·41-s + 4·43-s + 8·47-s + 49-s + 2·53-s − 12·59-s + 6·61-s + 6·65-s + 4·67-s − 12·71-s + 10·73-s + 8·79-s − 12·83-s + 2·85-s − 14·89-s + 6·91-s − 4·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 1.66·13-s + 0.485·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s − 1.11·29-s + 0.169·35-s + 0.986·37-s + 0.312·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.274·53-s − 1.56·59-s + 0.768·61-s + 0.744·65-s + 0.488·67-s − 1.42·71-s + 1.17·73-s + 0.900·79-s − 1.31·83-s + 0.216·85-s − 1.48·89-s + 0.628·91-s − 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.557230491\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.557230491\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.294114649649316099546000176408, −7.57083780041952707486232674545, −6.73564151664166741773491832288, −5.95469919480647266312415710566, −5.53562947695247287795876920358, −4.45453676145248847316790331476, −3.79862606631802281615681697334, −2.83592835194699145683966733921, −1.81509865059424392222288061858, −0.927684602942016600839548214425,
0.927684602942016600839548214425, 1.81509865059424392222288061858, 2.83592835194699145683966733921, 3.79862606631802281615681697334, 4.45453676145248847316790331476, 5.53562947695247287795876920358, 5.95469919480647266312415710566, 6.73564151664166741773491832288, 7.57083780041952707486232674545, 8.294114649649316099546000176408
