| L(s) = 1 | + 5-s + 13-s − 6·17-s + 8·19-s − 6·23-s + 25-s + 2·29-s − 6·31-s − 2·37-s − 10·41-s + 4·43-s + 4·47-s − 7·49-s + 4·53-s − 4·59-s − 2·61-s + 65-s + 10·67-s − 12·73-s − 4·83-s − 6·85-s − 14·89-s + 8·95-s − 4·97-s − 2·101-s − 16·103-s + 12·107-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.277·13-s − 1.45·17-s + 1.83·19-s − 1.25·23-s + 1/5·25-s + 0.371·29-s − 1.07·31-s − 0.328·37-s − 1.56·41-s + 0.609·43-s + 0.583·47-s − 49-s + 0.549·53-s − 0.520·59-s − 0.256·61-s + 0.124·65-s + 1.22·67-s − 1.40·73-s − 0.439·83-s − 0.650·85-s − 1.48·89-s + 0.820·95-s − 0.406·97-s − 0.199·101-s − 1.57·103-s + 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.23816521506155907032226292941, −6.76703319651270954296760015041, −5.92080275719295021607256970409, −5.40088206789605205594841702649, −4.61115587610486091105023429198, −3.81598080961701652340785873290, −3.02994646241461853060058096171, −2.12433087205259517291688919791, −1.34403557748428706930903751430, 0,
1.34403557748428706930903751430, 2.12433087205259517291688919791, 3.02994646241461853060058096171, 3.81598080961701652340785873290, 4.61115587610486091105023429198, 5.40088206789605205594841702649, 5.92080275719295021607256970409, 6.76703319651270954296760015041, 7.23816521506155907032226292941
