| L(s) = 1 | − 2·4-s − 2·9-s + 8·11-s + 3·16-s + 4·19-s + 4·31-s + 4·36-s + 16·41-s − 16·44-s + 20·49-s + 12·59-s + 8·61-s − 4·64-s + 28·71-s − 8·76-s + 36·79-s + 3·81-s + 12·89-s − 16·99-s + 44·101-s + 32·109-s + 2·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 4-s − 2/3·9-s + 2.41·11-s + 3/4·16-s + 0.917·19-s + 0.718·31-s + 2/3·36-s + 2.49·41-s − 2.41·44-s + 20/7·49-s + 1.56·59-s + 1.02·61-s − 1/2·64-s + 3.32·71-s − 0.917·76-s + 4.05·79-s + 1/3·81-s + 1.27·89-s − 1.60·99-s + 4.37·101-s + 3.06·109-s + 2/11·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.416324079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.416324079\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 186 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 23 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 810 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 506 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T^{2} + 875 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 55 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 2 T + 51 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 5226 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 103 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 T + 51 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 254 T^{2} + 25059 T^{4} - 254 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 14 T + 164 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 266 T^{2} + 28299 T^{4} - 266 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 18 T + 227 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 318 T^{2} + 39011 T^{4} - 318 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 14970 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.31074215802531489618822421755, −6.01718902742684402155845326799, −5.83406393139373014800194033839, −5.49802935113631877818317993759, −5.42151456533695700342534489877, −5.27875310449029642461354347961, −5.08425831493630026748151401532, −4.61296657343217588125430451159, −4.58112555628060699119612630770, −4.38147611487047617980225135059, −3.99477473172513420733561292919, −3.97931573118379837891004225629, −3.82917997625488571070364715563, −3.48233844102926234694035031009, −3.29402079869067091185337139276, −3.01974169640940254419882780082, −3.00402867640629292525376570369, −2.25869142473229097200356071145, −2.14941746668255282399398898964, −1.98766430306102036390346906195, −1.94022812470502664695990578927, −0.890269677368230293234200310857, −0.828525079832537210064425432570, −0.809795960787126667437717395981, −0.76063953128108723011741886526,
0.76063953128108723011741886526, 0.809795960787126667437717395981, 0.828525079832537210064425432570, 0.890269677368230293234200310857, 1.94022812470502664695990578927, 1.98766430306102036390346906195, 2.14941746668255282399398898964, 2.25869142473229097200356071145, 3.00402867640629292525376570369, 3.01974169640940254419882780082, 3.29402079869067091185337139276, 3.48233844102926234694035031009, 3.82917997625488571070364715563, 3.97931573118379837891004225629, 3.99477473172513420733561292919, 4.38147611487047617980225135059, 4.58112555628060699119612630770, 4.61296657343217588125430451159, 5.08425831493630026748151401532, 5.27875310449029642461354347961, 5.42151456533695700342534489877, 5.49802935113631877818317993759, 5.83406393139373014800194033839, 6.01718902742684402155845326799, 6.31074215802531489618822421755
