| L(s) = 1 | + 3-s + 5-s − 2·7-s + 9-s + 4·11-s − 4·13-s + 15-s − 6·17-s − 2·21-s + 25-s + 27-s + 4·29-s − 31-s + 4·33-s − 2·35-s + 4·37-s − 4·39-s − 6·41-s + 8·43-s + 45-s + 12·47-s − 3·49-s − 6·51-s − 2·53-s + 4·55-s − 6·59-s + 10·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 1.20·11-s − 1.10·13-s + 0.258·15-s − 1.45·17-s − 0.436·21-s + 1/5·25-s + 0.192·27-s + 0.742·29-s − 0.179·31-s + 0.696·33-s − 0.338·35-s + 0.657·37-s − 0.640·39-s − 0.937·41-s + 1.21·43-s + 0.149·45-s + 1.75·47-s − 3/7·49-s − 0.840·51-s − 0.274·53-s + 0.539·55-s − 0.781·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.413867245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.413867245\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−7.925103411201630563889412624958, −6.97345380034827028515170513579, −6.69587345641027245879446645844, −5.97060914346962855475496103984, −4.95494489886708548001303315052, −4.26011497356035344902388089394, −3.53850977057990767312666552761, −2.56901384875670560975924580197, −2.04129144388304018165427341836, −0.74438243992279157409105415040,
0.74438243992279157409105415040, 2.04129144388304018165427341836, 2.56901384875670560975924580197, 3.53850977057990767312666552761, 4.26011497356035344902388089394, 4.95494489886708548001303315052, 5.97060914346962855475496103984, 6.69587345641027245879446645844, 6.97345380034827028515170513579, 7.925103411201630563889412624958
