L(s) = 1 | − 5-s − 2·11-s + 6·17-s − 2·19-s − 3·23-s − 4·29-s + 5·31-s + 20·37-s − 6·41-s + 6·43-s − 8·47-s + 7·49-s + 6·53-s + 2·55-s − 5·61-s + 2·67-s + 4·71-s + 12·73-s + 11·79-s − 9·83-s − 6·85-s + 20·89-s + 2·95-s − 8·97-s + 12·101-s + 12·103-s − 8·107-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.603·11-s + 1.45·17-s − 0.458·19-s − 0.625·23-s − 0.742·29-s + 0.898·31-s + 3.28·37-s − 0.937·41-s + 0.914·43-s − 1.16·47-s + 49-s + 0.824·53-s + 0.269·55-s − 0.640·61-s + 0.244·67-s + 0.474·71-s + 1.40·73-s + 1.23·79-s − 0.987·83-s − 0.650·85-s + 2.11·89-s + 0.205·95-s − 0.812·97-s + 1.19·101-s + 1.18·103-s − 0.773·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.610463609\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.610463609\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
good | 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 6 T - 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.699157407487801783117826202595, −8.375644653711369405055821968656, −7.949927499266847783744645699148, −7.895713173417943716693383897678, −7.35312361971692527748573688796, −7.21259472537554617289025941703, −6.43557297735975774945411042782, −6.25002722702370821511111696866, −5.73163655044939700886347902003, −5.61619642531035342431065107657, −4.82008112447506116388268978524, −4.73146679433286267611047752682, −4.10288208409951118958109915722, −3.82710629090034865094885928173, −3.10984295732141777961756380764, −3.04917901351017251881082674340, −2.09177287399667536600513909418, −2.07715463233027668175503528762, −0.885367002401265269664088565825, −0.65920071246210068966079514178,
0.65920071246210068966079514178, 0.885367002401265269664088565825, 2.07715463233027668175503528762, 2.09177287399667536600513909418, 3.04917901351017251881082674340, 3.10984295732141777961756380764, 3.82710629090034865094885928173, 4.10288208409951118958109915722, 4.73146679433286267611047752682, 4.82008112447506116388268978524, 5.61619642531035342431065107657, 5.73163655044939700886347902003, 6.25002722702370821511111696866, 6.43557297735975774945411042782, 7.21259472537554617289025941703, 7.35312361971692527748573688796, 7.895713173417943716693383897678, 7.949927499266847783744645699148, 8.375644653711369405055821968656, 8.699157407487801783117826202595