| L(s) = 1 | − 2·3-s + 4·5-s + 4·7-s + 2·9-s + 8·13-s − 8·15-s + 8·17-s + 8·19-s − 8·21-s + 2·25-s − 6·27-s + 16·35-s − 24·37-s − 16·39-s + 8·45-s − 8·49-s − 16·51-s − 16·57-s + 8·63-s + 32·65-s − 4·75-s + 11·81-s + 12·83-s + 32·85-s + 32·91-s + 32·95-s − 32·101-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s + 1.51·7-s + 2/3·9-s + 2.21·13-s − 2.06·15-s + 1.94·17-s + 1.83·19-s − 1.74·21-s + 2/5·25-s − 1.15·27-s + 2.70·35-s − 3.94·37-s − 2.56·39-s + 1.19·45-s − 8/7·49-s − 2.24·51-s − 2.11·57-s + 1.00·63-s + 3.96·65-s − 0.461·75-s + 11/9·81-s + 1.31·83-s + 3.47·85-s + 3.35·91-s + 3.28·95-s − 3.18·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.125365939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.125365939\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| good | 7 | $D_{4}$ | \( ( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 2638 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_4\times C_2$ | \( 1 - 76 T^{2} + 3046 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 64 T^{2} + 3742 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 5038 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 2998 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 8574 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 8038 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 204 T^{2} + 20006 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 114 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 196 T^{2} + 23302 T^{4} - 196 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.35457188956882177201292349012, −6.35293877173931294965281898595, −6.02384617514785693454845659586, −5.73954561477660369910692330548, −5.70866049226917336200488860738, −5.51529630432656155778138240866, −5.34683142609885895839023787148, −5.27272636591011672547538204911, −4.90236504855492429881050590042, −4.81574271665069569587958182192, −4.52903679137464369762874948625, −4.26722089127849816611634357586, −3.65496844501323009813583875261, −3.55012628166787667464948530040, −3.54531824839118269724297477756, −3.47019601178675685245474746040, −2.89141154897237030512165251806, −2.72125564515835117949195986429, −2.02575835566290647547803316759, −1.99694225793644290427788135479, −1.56543519787092597308843189020, −1.53909520135487834531496044688, −1.36325112006812944098780144196, −0.906206098396523172788130310920, −0.44934588871415829609363851441,
0.44934588871415829609363851441, 0.906206098396523172788130310920, 1.36325112006812944098780144196, 1.53909520135487834531496044688, 1.56543519787092597308843189020, 1.99694225793644290427788135479, 2.02575835566290647547803316759, 2.72125564515835117949195986429, 2.89141154897237030512165251806, 3.47019601178675685245474746040, 3.54531824839118269724297477756, 3.55012628166787667464948530040, 3.65496844501323009813583875261, 4.26722089127849816611634357586, 4.52903679137464369762874948625, 4.81574271665069569587958182192, 4.90236504855492429881050590042, 5.27272636591011672547538204911, 5.34683142609885895839023787148, 5.51529630432656155778138240866, 5.70866049226917336200488860738, 5.73954561477660369910692330548, 6.02384617514785693454845659586, 6.35293877173931294965281898595, 6.35457188956882177201292349012
