| L(s) = 1 | − 4-s − 3·16-s + 2·17-s + 8·23-s + 7·25-s − 2·29-s − 10·43-s + 15·49-s − 22·53-s + 32·61-s + 3·64-s − 2·68-s + 30·79-s − 8·92-s − 7·100-s − 22·101-s + 22·103-s + 18·113-s + 2·116-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 3/4·16-s + 0.485·17-s + 1.66·23-s + 7/5·25-s − 0.371·29-s − 1.52·43-s + 15/7·49-s − 3.02·53-s + 4.09·61-s + 3/8·64-s − 0.242·68-s + 3.37·79-s − 0.834·92-s − 0.699·100-s − 2.18·101-s + 2.16·103-s + 1.69·113-s + 0.185·116-s + 3.27·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.743394382\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.743394382\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + T^{2} + p^{2} T^{4} + p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^3$ | \( 1 - 7 T^{2} + 24 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 15 T^{2} + 116 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 24 T^{2} + 254 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 29 | $D_{4}$ | \( ( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 115 T^{2} + 5224 T^{4} - 115 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 79 T^{2} + 3784 T^{4} - 79 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 155 T^{2} + 9364 T^{4} - 155 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 5 T + 88 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 11 T + 98 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 104 T^{2} + 6334 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 16 T + 169 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 247 T^{2} + 24124 T^{4} - 247 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 186 T^{2} + 16859 T^{4} - 186 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 15 T + 210 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 248 T^{2} + 27454 T^{4} - 248 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 160 T^{2} + 16734 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 295 T^{2} + 39856 T^{4} - 295 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.73343977855231808573855731968, −6.60135736738156978987578188732, −6.53188151316221986926612149109, −6.06467410282436298617906518941, −5.94040065779220960938064780560, −5.43106179969596408526065099035, −5.39244292840511893873551545974, −5.32425178048999822647102106585, −4.87111174822761321620364046867, −4.84936111500598726048211830937, −4.67052590880792840773830734182, −4.34974275116811973382139400763, −3.98139402152287830611797281540, −3.83474350001262017664087974199, −3.62048825687351226800046125626, −3.20798880525452539633030946132, −3.09301942385086955850903871728, −2.86683408246551183046732003024, −2.56165624439918345731330092105, −2.02155372941901206529855282017, −2.00553202849314730377912392907, −1.64666864131275140311928411999, −1.01863710236056831658686766935, −0.73891486913757544781750861954, −0.52022721083876749509999785499,
0.52022721083876749509999785499, 0.73891486913757544781750861954, 1.01863710236056831658686766935, 1.64666864131275140311928411999, 2.00553202849314730377912392907, 2.02155372941901206529855282017, 2.56165624439918345731330092105, 2.86683408246551183046732003024, 3.09301942385086955850903871728, 3.20798880525452539633030946132, 3.62048825687351226800046125626, 3.83474350001262017664087974199, 3.98139402152287830611797281540, 4.34974275116811973382139400763, 4.67052590880792840773830734182, 4.84936111500598726048211830937, 4.87111174822761321620364046867, 5.32425178048999822647102106585, 5.39244292840511893873551545974, 5.43106179969596408526065099035, 5.94040065779220960938064780560, 6.06467410282436298617906518941, 6.53188151316221986926612149109, 6.60135736738156978987578188732, 6.73343977855231808573855731968
