| L(s) = 1 | + 4-s + 3·5-s − 6·11-s + 16-s + 4·19-s + 3·20-s + 4·25-s + 12·29-s + 10·31-s − 12·41-s − 6·44-s − 13·49-s − 18·55-s + 24·59-s + 16·61-s + 64-s + 4·76-s + 16·79-s + 3·80-s − 36·89-s + 12·95-s + 4·100-s − 6·101-s + 4·109-s + 12·116-s + 5·121-s + 10·124-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.34·5-s − 1.80·11-s + 1/4·16-s + 0.917·19-s + 0.670·20-s + 4/5·25-s + 2.22·29-s + 1.79·31-s − 1.87·41-s − 0.904·44-s − 1.85·49-s − 2.42·55-s + 3.12·59-s + 2.04·61-s + 1/8·64-s + 0.458·76-s + 1.80·79-s + 0.335·80-s − 3.81·89-s + 1.23·95-s + 2/5·100-s − 0.597·101-s + 0.383·109-s + 1.11·116-s + 5/11·121-s + 0.898·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.981095907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.981095907\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−9.872045258839871807563899217385, −9.726216919693265926792364692853, −8.592975668443455053869237962204, −8.268048909391662018895990050852, −8.018239560748665330030327379373, −7.02301072665066136063726833931, −6.73860900674537002760989556093, −6.23182947846711894237454668645, −5.37363015537505503550836847886, −5.27556992594019260057284686075, −4.64652891291450929816238213682, −3.48322127537334094984127398692, −2.62987626086243772212633915702, −2.44850978333263328730546301919, −1.21757578202308075265137076550,
1.21757578202308075265137076550, 2.44850978333263328730546301919, 2.62987626086243772212633915702, 3.48322127537334094984127398692, 4.64652891291450929816238213682, 5.27556992594019260057284686075, 5.37363015537505503550836847886, 6.23182947846711894237454668645, 6.73860900674537002760989556093, 7.02301072665066136063726833931, 8.018239560748665330030327379373, 8.268048909391662018895990050852, 8.592975668443455053869237962204, 9.726216919693265926792364692853, 9.872045258839871807563899217385
