| L(s) = 1 | − 2·4-s − 2·9-s + 12·11-s + 3·16-s + 20·19-s − 10·25-s − 16·29-s + 8·31-s + 4·36-s − 24·44-s + 20·49-s + 8·59-s + 12·61-s − 4·64-s + 8·71-s − 40·76-s − 8·79-s + 3·81-s + 48·89-s − 24·99-s + 20·100-s − 28·101-s + 12·109-s + 32·116-s + 56·121-s − 16·124-s + 127-s + ⋯ |
| L(s) = 1 | − 4-s − 2/3·9-s + 3.61·11-s + 3/4·16-s + 4.58·19-s − 2·25-s − 2.97·29-s + 1.43·31-s + 2/3·36-s − 3.61·44-s + 20/7·49-s + 1.04·59-s + 1.53·61-s − 1/2·64-s + 0.949·71-s − 4.58·76-s − 0.900·79-s + 1/3·81-s + 5.08·89-s − 2.41·99-s + 2·100-s − 2.78·101-s + 1.14·109-s + 2.97·116-s + 5.09·121-s − 1.43·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.586452026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.586452026\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_4$ | \( ( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 31 | $C_4$ | \( ( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 1718 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 4398 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 982 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 6 T + 126 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 12182 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_4$ | \( ( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 180 T^{2} + 15878 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 4 T + 142 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 36 T^{2} + 11222 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 24 T + 302 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 120 T^{2} + 17918 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.20950372246305573815087607833, −6.70636719900031347112568444633, −6.61607485360118670497293756824, −6.28998072184615532373585326525, −6.24021152893788896918988029078, −5.82620419311320945768086043482, −5.71237488003820205320364543167, −5.36088114769133888795645993317, −5.26430223891565387563656588570, −5.08589184260348488999640549100, −5.00626880318717505351841521580, −4.24886825253269357113393462942, −4.05903672584067963898766658599, −3.96910525310792983259091587082, −3.86828960203887803483098214347, −3.56692118234691918471405970566, −3.42314656851286579869555511541, −3.11770325065176283827982959646, −2.63226795511714093796588274950, −2.38123209585988732735866482896, −1.88940493153394150275301177940, −1.45657001153063217589613621064, −1.31448181767953577256936909241, −0.78334332832706153468519125823, −0.67698037312515517408818103287,
0.67698037312515517408818103287, 0.78334332832706153468519125823, 1.31448181767953577256936909241, 1.45657001153063217589613621064, 1.88940493153394150275301177940, 2.38123209585988732735866482896, 2.63226795511714093796588274950, 3.11770325065176283827982959646, 3.42314656851286579869555511541, 3.56692118234691918471405970566, 3.86828960203887803483098214347, 3.96910525310792983259091587082, 4.05903672584067963898766658599, 4.24886825253269357113393462942, 5.00626880318717505351841521580, 5.08589184260348488999640549100, 5.26430223891565387563656588570, 5.36088114769133888795645993317, 5.71237488003820205320364543167, 5.82620419311320945768086043482, 6.24021152893788896918988029078, 6.28998072184615532373585326525, 6.61607485360118670497293756824, 6.70636719900031347112568444633, 7.20950372246305573815087607833
