| L(s) = 1 | − 2·4-s + 4·5-s + 3·16-s − 8·20-s + 2·25-s − 12·29-s − 16·31-s − 16·41-s + 2·49-s − 24·59-s − 4·64-s + 24·71-s + 32·79-s + 12·80-s − 24·89-s − 4·100-s − 16·101-s − 16·109-s + 24·116-s − 38·121-s + 32·124-s − 28·125-s + 127-s + 131-s + 137-s + 139-s − 48·145-s + ⋯ |
| L(s) = 1 | − 4-s + 1.78·5-s + 3/4·16-s − 1.78·20-s + 2/5·25-s − 2.22·29-s − 2.87·31-s − 2.49·41-s + 2/7·49-s − 3.12·59-s − 1/2·64-s + 2.84·71-s + 3.60·79-s + 1.34·80-s − 2.54·89-s − 2/5·100-s − 1.59·101-s − 1.53·109-s + 2.22·116-s − 3.45·121-s + 2.87·124-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.98·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6315233954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6315233954\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 2 T^{2} + 51 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 26 T^{2} + 315 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 1766 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 75 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 8 T + 71 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 110 T^{2} + 6291 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 1670 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 198 T^{2} + 15371 T^{4} - 198 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 12 T + 142 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 95 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 22710 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 4230 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 16 T + 210 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 25974 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 12 T + 202 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 106 T^{2} + 17739 T^{4} + 106 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.05928591248729746335770480092, −5.93059053448217258599807807789, −5.67257457832960688173897097586, −5.47723423397477909049701796446, −5.28630625415500912623573924897, −5.15492589213152165324813670544, −4.89929883081616996000506065302, −4.86989690094069084193389239549, −4.73736199799376563421740348383, −3.96589255552770777008682615996, −3.90971570581466493456011738165, −3.88872124229261233386162952282, −3.81074318996677028326851004057, −3.51790202187807774132992146549, −3.14902266570745522633527318623, −2.92008142746491852219870801691, −2.65079663441318087907471452786, −2.27986452984781918195319751453, −2.18028046424781190981383101556, −1.74187625480374322789958271513, −1.60803541460927372047435980271, −1.57522249214164407597717036479, −1.20530261493963759217413866081, −0.48647773820450725594181615798, −0.15146200913530119847390679484,
0.15146200913530119847390679484, 0.48647773820450725594181615798, 1.20530261493963759217413866081, 1.57522249214164407597717036479, 1.60803541460927372047435980271, 1.74187625480374322789958271513, 2.18028046424781190981383101556, 2.27986452984781918195319751453, 2.65079663441318087907471452786, 2.92008142746491852219870801691, 3.14902266570745522633527318623, 3.51790202187807774132992146549, 3.81074318996677028326851004057, 3.88872124229261233386162952282, 3.90971570581466493456011738165, 3.96589255552770777008682615996, 4.73736199799376563421740348383, 4.86989690094069084193389239549, 4.89929883081616996000506065302, 5.15492589213152165324813670544, 5.28630625415500912623573924897, 5.47723423397477909049701796446, 5.67257457832960688173897097586, 5.93059053448217258599807807789, 6.05928591248729746335770480092
