| L(s) = 1 | − 4-s − 2·11-s − 3·16-s + 12·19-s + 2·29-s − 4·41-s + 2·44-s − 2·49-s − 16·59-s + 12·61-s + 3·64-s − 32·71-s − 12·76-s + 18·79-s + 12·89-s + 32·101-s − 46·109-s − 2·116-s − 33·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.603·11-s − 3/4·16-s + 2.75·19-s + 0.371·29-s − 0.624·41-s + 0.301·44-s − 2/7·49-s − 2.08·59-s + 1.53·61-s + 3/8·64-s − 3.79·71-s − 1.37·76-s + 2.02·79-s + 1.27·89-s + 3.18·101-s − 4.40·109-s − 0.185·116-s − 3·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.529328511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.529328511\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + T^{2} + p^{2} T^{4} + p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 31 T^{2} + 472 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 47 T^{2} + 1024 T^{4} - 47 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 56 T^{2} + 1774 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 2 T + 66 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 3934 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 99 T^{2} + 5912 T^{4} - 99 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 176 T^{2} + 13294 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 - 6 T - 22 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 11734 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 10150 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 9 T + 174 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 6 T + 170 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 135 T^{2} + 14768 T^{4} - 135 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.71644333305596354895716346394, −6.60600736009605813818825208620, −6.50401882888075287433743111109, −5.86960166572514984181997014938, −5.78480150654248687035572992558, −5.62999699156149640233807671369, −5.40117648155390010991853651392, −5.20199900037877908056199164458, −5.08933634328711145828021610982, −4.65637395094068052694216196743, −4.51626160158364359327914286232, −4.31392834042464665713594538837, −4.23148591633908601008736655222, −3.65525108501171721566247332147, −3.58267958984426480832226036404, −3.20212164108774119709264941368, −2.90056083926963675153801007478, −2.88872793543652654702833825380, −2.77709823792998055210940396600, −1.89732271496415903857570524637, −1.88905477895534508055376154293, −1.74880183530616904935616320243, −1.06934631818962637403639134847, −0.74958019935255402016936242758, −0.39029540564452322454680269083,
0.39029540564452322454680269083, 0.74958019935255402016936242758, 1.06934631818962637403639134847, 1.74880183530616904935616320243, 1.88905477895534508055376154293, 1.89732271496415903857570524637, 2.77709823792998055210940396600, 2.88872793543652654702833825380, 2.90056083926963675153801007478, 3.20212164108774119709264941368, 3.58267958984426480832226036404, 3.65525108501171721566247332147, 4.23148591633908601008736655222, 4.31392834042464665713594538837, 4.51626160158364359327914286232, 4.65637395094068052694216196743, 5.08933634328711145828021610982, 5.20199900037877908056199164458, 5.40117648155390010991853651392, 5.62999699156149640233807671369, 5.78480150654248687035572992558, 5.86960166572514984181997014938, 6.50401882888075287433743111109, 6.60600736009605813818825208620, 6.71644333305596354895716346394
