| L(s) = 1 | + 4·4-s + 2·7-s + 5·16-s − 12·19-s + 32·25-s + 8·28-s + 24·31-s + 12·37-s − 7·49-s + 2·64-s − 48·76-s + 128·100-s − 92·103-s + 10·112-s + 80·121-s + 96·124-s + 127-s + 131-s − 24·133-s + 137-s + 139-s + 48·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 2·4-s + 0.755·7-s + 5/4·16-s − 2.75·19-s + 32/5·25-s + 1.51·28-s + 4.31·31-s + 1.97·37-s − 49-s + 1/4·64-s − 5.50·76-s + 64/5·100-s − 9.06·103-s + 0.944·112-s + 7.27·121-s + 8.62·124-s + 0.0887·127-s + 0.0873·131-s − 2.08·133-s + 0.0854·137-s + 0.0848·139-s + 3.94·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{36} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{36} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.475021460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.475021460\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T^{2} + 11 T^{4} - 13 p T^{6} + 11 p^{2} T^{8} - p^{6} T^{10} + p^{6} T^{12} \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 - T + 5 T^{2} - 2 T^{3} + 5 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| good | 5 | \( ( 1 - 16 T^{2} + 154 T^{4} - 918 T^{6} + 154 p^{2} T^{8} - 16 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 11 | \( ( 1 - 40 T^{2} + 722 T^{4} - 8858 T^{6} + 722 p^{2} T^{8} - 40 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 13 | \( ( 1 - 23 T^{2} + 331 T^{4} - 3102 T^{6} + 331 p^{2} T^{8} - 23 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 17 | \( ( 1 - 50 T^{2} + 1419 T^{4} - 28884 T^{6} + 1419 p^{2} T^{8} - 50 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 19 | \( ( 1 + 3 T + 46 T^{2} + 117 T^{3} + 46 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 23 | \( ( 1 - 112 T^{2} + 5594 T^{4} - 163154 T^{6} + 5594 p^{2} T^{8} - 112 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 29 | \( ( 1 + 42 T^{2} + 1155 T^{4} + 14916 T^{6} + 1155 p^{2} T^{8} + 42 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 31 | \( ( 1 - 6 T + 48 T^{2} - 264 T^{3} + 48 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 37 | \( ( 1 - 3 T + 58 T^{2} - 39 T^{3} + 58 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 41 | \( ( 1 - 112 T^{2} + 5698 T^{4} - 227382 T^{6} + 5698 p^{2} T^{8} - 112 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 43 | \( ( 1 - 154 T^{2} + 11947 T^{4} - 616020 T^{6} + 11947 p^{2} T^{8} - 154 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 47 | \( ( 1 + 102 T^{2} + 9163 T^{4} + 465404 T^{6} + 9163 p^{2} T^{8} + 102 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 53 | \( ( 1 + 86 T^{2} + 7463 T^{4} + 484852 T^{6} + 7463 p^{2} T^{8} + 86 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 59 | \( ( 1 - 74 T^{2} + 12051 T^{4} - 8876 p T^{6} + 12051 p^{2} T^{8} - 74 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 61 | \( ( 1 - 155 T^{2} + 12259 T^{4} - 719586 T^{6} + 12259 p^{2} T^{8} - 155 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 67 | \( ( 1 - 347 T^{2} + 53251 T^{4} - 4619670 T^{6} + 53251 p^{2} T^{8} - 347 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 71 | \( ( 1 - 276 T^{2} + 38818 T^{4} - 3386178 T^{6} + 38818 p^{2} T^{8} - 276 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 73 | \( ( 1 - 247 T^{2} + 30171 T^{4} - 2531358 T^{6} + 30171 p^{2} T^{8} - 247 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 79 | \( ( 1 - 59 T^{2} + 13987 T^{4} - 673302 T^{6} + 13987 p^{2} T^{8} - 59 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 83 | \( ( 1 + 102 T^{2} + 16579 T^{4} + 1402844 T^{6} + 16579 p^{2} T^{8} + 102 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 89 | \( ( 1 - 88 T^{2} + 17938 T^{4} - 1434078 T^{6} + 17938 p^{2} T^{8} - 88 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 97 | \( ( 1 - 371 T^{2} + 67195 T^{4} - 7803234 T^{6} + 67195 p^{2} T^{8} - 371 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.27359924372681644254322528290, −3.12798842061964465775359835329, −3.05039478781326385391011308621, −2.99529756859197918555491438099, −2.90606093386302356053304316954, −2.81570609734148008068316004894, −2.67653861693477492690961801451, −2.62163008687184372534546505080, −2.55819064681312919507991744614, −2.55308847329954753259605855763, −2.42373456696801163945628271283, −2.19154746056676662454559177112, −2.19104322573828698505689831265, −1.90474528797083608149634715383, −1.79587616647147353040132944317, −1.73967638539524123511930909092, −1.66063184999029296929821447947, −1.52534517611538518383137729204, −1.32906954304183702859024144923, −1.04565973734058371634327146156, −0.980621634981862510941683382256, −0.831311916579621776589952100610, −0.798657005389844421301671526634, −0.73977614969418760020869790080, −0.10267157456270152388171249941,
0.10267157456270152388171249941, 0.73977614969418760020869790080, 0.798657005389844421301671526634, 0.831311916579621776589952100610, 0.980621634981862510941683382256, 1.04565973734058371634327146156, 1.32906954304183702859024144923, 1.52534517611538518383137729204, 1.66063184999029296929821447947, 1.73967638539524123511930909092, 1.79587616647147353040132944317, 1.90474528797083608149634715383, 2.19104322573828698505689831265, 2.19154746056676662454559177112, 2.42373456696801163945628271283, 2.55308847329954753259605855763, 2.55819064681312919507991744614, 2.62163008687184372534546505080, 2.67653861693477492690961801451, 2.81570609734148008068316004894, 2.90606093386302356053304316954, 2.99529756859197918555491438099, 3.05039478781326385391011308621, 3.12798842061964465775359835329, 3.27359924372681644254322528290
Plot not available for L-functions of degree greater than 10.
