| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s + 8-s + 9-s − 2·12-s + 4·13-s + 16-s + 18-s − 4·19-s − 2·24-s + 4·26-s + 4·27-s + 6·29-s + 10·31-s + 32-s + 36-s − 2·37-s − 4·38-s − 8·39-s − 12·41-s − 4·43-s + 6·47-s − 2·48-s + 4·52-s + 6·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.353·8-s + 1/3·9-s − 0.577·12-s + 1.10·13-s + 1/4·16-s + 0.235·18-s − 0.917·19-s − 0.408·24-s + 0.784·26-s + 0.769·27-s + 1.11·29-s + 1.79·31-s + 0.176·32-s + 1/6·36-s − 0.328·37-s − 0.648·38-s − 1.28·39-s − 1.87·41-s − 0.609·43-s + 0.875·47-s − 0.288·48-s + 0.554·52-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.761628232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.761628232\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.59196848867361, −12.16343688553134, −11.85730603192520, −11.35763320701213, −11.03062600954144, −10.55843378059465, −10.01987558968630, −9.896968048846881, −8.712704873830970, −8.507104124718764, −8.248289357067971, −7.319425166753076, −6.769571644867161, −6.499947641505361, −6.124423399038275, −5.559971783980313, −5.173701601755703, −4.588239103707259, −4.228760884007340, −3.602358010339070, −3.001457314081674, −2.467779341987041, −1.690889810512247, −1.063219899143116, −0.4722773468784139,
0.4722773468784139, 1.063219899143116, 1.690889810512247, 2.467779341987041, 3.001457314081674, 3.602358010339070, 4.228760884007340, 4.588239103707259, 5.173701601755703, 5.559971783980313, 6.124423399038275, 6.499947641505361, 6.769571644867161, 7.319425166753076, 8.248289357067971, 8.507104124718764, 8.712704873830970, 9.896968048846881, 10.01987558968630, 10.55843378059465, 11.03062600954144, 11.35763320701213, 11.85730603192520, 12.16343688553134, 12.59196848867361
