| L(s) = 1 | − 2·4-s + 4·5-s + 3·16-s + 8·19-s − 8·20-s + 8·25-s − 20·29-s + 4·31-s − 2·49-s − 20·59-s + 24·61-s − 4·64-s − 12·71-s − 16·76-s + 16·79-s + 12·80-s − 44·89-s + 32·95-s − 16·100-s − 8·109-s + 40·116-s − 32·121-s − 8·124-s + 20·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 4-s + 1.78·5-s + 3/4·16-s + 1.83·19-s − 1.78·20-s + 8/5·25-s − 3.71·29-s + 0.718·31-s − 2/7·49-s − 2.60·59-s + 3.07·61-s − 1/2·64-s − 1.42·71-s − 1.83·76-s + 1.80·79-s + 1.34·80-s − 4.66·89-s + 3.28·95-s − 8/5·100-s − 0.766·109-s + 3.71·116-s − 2.90·121-s − 0.718·124-s + 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1944670996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1944670996\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 11 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 38 T^{2} + 675 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 33 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 4 T + 36 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 78 T^{2} + 2555 T^{4} - 78 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 10 T + 77 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 2 T + 57 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 8 T^{2} - 702 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 122 T^{2} + 7323 T^{4} - 122 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 12 T^{2} + 2918 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 182 T^{2} + 13683 T^{4} - 182 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 10 T + 119 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 109 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 6 T + 145 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 11238 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 8 T + 168 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 212 T^{2} + 21558 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 17298 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.50271222478550373176498432703, −6.35065872767443544737753622833, −6.15120762046518286313136996523, −5.65312625010609610689390202920, −5.61925965456097503966054943385, −5.54036308206768014627117858205, −5.45208048257436306173383321837, −5.09404814431590079050007073890, −5.00138710951817462182885749445, −4.70788701777757320643217634608, −4.45319351084419590248410426917, −4.02704411157205392458795156460, −3.89693893103865129033136995793, −3.69687116996535708115852279001, −3.66300761620991692243727478484, −3.01966594615383454938022082614, −2.93147862329681896366019937112, −2.71851980375902188220281637257, −2.45440817029856750982392414213, −1.90396670393004707006624412406, −1.87177214449218186984493102303, −1.43380870849846408189054225702, −1.25696131815046017018142111365, −0.916072889152521710891946943810, −0.07501489018377843716941115833,
0.07501489018377843716941115833, 0.916072889152521710891946943810, 1.25696131815046017018142111365, 1.43380870849846408189054225702, 1.87177214449218186984493102303, 1.90396670393004707006624412406, 2.45440817029856750982392414213, 2.71851980375902188220281637257, 2.93147862329681896366019937112, 3.01966594615383454938022082614, 3.66300761620991692243727478484, 3.69687116996535708115852279001, 3.89693893103865129033136995793, 4.02704411157205392458795156460, 4.45319351084419590248410426917, 4.70788701777757320643217634608, 5.00138710951817462182885749445, 5.09404814431590079050007073890, 5.45208048257436306173383321837, 5.54036308206768014627117858205, 5.61925965456097503966054943385, 5.65312625010609610689390202920, 6.15120762046518286313136996523, 6.35065872767443544737753622833, 6.50271222478550373176498432703
