| L(s) = 1 | + 9·9-s − 2·17-s + 2·23-s + 17·25-s − 10·31-s + 4·41-s − 30·47-s − 16·71-s − 10·73-s − 22·79-s + 35·81-s − 10·89-s − 20·97-s + 46·103-s + 12·113-s + 29·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 18·153-s + 157-s + 163-s + 167-s + 46·169-s + ⋯ |
| L(s) = 1 | + 3·9-s − 0.485·17-s + 0.417·23-s + 17/5·25-s − 1.79·31-s + 0.624·41-s − 4.37·47-s − 1.89·71-s − 1.17·73-s − 2.47·79-s + 35/9·81-s − 1.05·89-s − 2.03·97-s + 4.53·103-s + 1.12·113-s + 2.63·121-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 − 1.45·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.53·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.412821693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.412821693\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - p^{2} T^{2} + 46 T^{4} - 161 T^{6} + 46 p^{2} T^{8} - p^{6} T^{10} + p^{6} T^{12} \) |
| 5 | \( 1 - 17 T^{2} + 142 T^{4} - 813 T^{6} + 142 p^{2} T^{8} - 17 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 29 T^{2} + 514 T^{4} - 555 p T^{6} + 514 p^{2} T^{8} - 29 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( 1 - 46 T^{2} + 1191 T^{4} - 18804 T^{6} + 1191 p^{2} T^{8} - 46 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 + T + 2 p T^{2} + 13 T^{3} + 2 p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( 1 - 53 T^{2} + 1594 T^{4} - 32937 T^{6} + 1594 p^{2} T^{8} - 53 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 - T + 62 T^{2} - 37 T^{3} + 62 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( 1 - 102 T^{2} + 4567 T^{4} - 141988 T^{6} + 4567 p^{2} T^{8} - 102 p^{4} T^{10} + p^{6} T^{12} \) |
| 31 | \( ( 1 + 5 T + 96 T^{2} + 303 T^{3} + 96 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 161 T^{2} + 12406 T^{4} - 578469 T^{6} + 12406 p^{2} T^{8} - 161 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 2 T + 83 T^{2} - 80 T^{3} + 83 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 94 T^{2} + 3543 T^{4} - 112068 T^{6} + 3543 p^{2} T^{8} - 94 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( ( 1 + 15 T + 168 T^{2} + 1221 T^{3} + 168 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( 1 - 161 T^{2} + 13750 T^{4} - 824709 T^{6} + 13750 p^{2} T^{8} - 161 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( 1 - 3 p T^{2} + 14590 T^{4} - 874057 T^{6} + 14590 p^{2} T^{8} - 3 p^{5} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 - 73 T^{2} + 9270 T^{4} - 398973 T^{6} + 9270 p^{2} T^{8} - 73 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( 1 - 361 T^{2} + 56598 T^{4} - 4944537 T^{6} + 56598 p^{2} T^{8} - 361 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 8 T + 157 T^{2} + 704 T^{3} + 157 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 + 5 T + 126 T^{2} + 289 T^{3} + 126 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( ( 1 + 11 T + 258 T^{2} + 1747 T^{3} + 258 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 446 T^{2} + 86503 T^{4} - 9357636 T^{6} + 86503 p^{2} T^{8} - 446 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 5 T + 158 T^{2} + 1121 T^{3} + 158 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( ( 1 + 10 T + 255 T^{2} + 1968 T^{3} + 255 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.88488207170023570045902653561, −4.69375402562755934594745210233, −4.47792951589070623289327378626, −4.46588215099134614360666220936, −4.40611057910497230908842718511, −4.32147603313826274834685607093, −4.21443030537951812119616274626, −3.79227115410962113396518390830, −3.60229699204304422184199107173, −3.58903015924606180163727973865, −3.24552442328863808924308647627, −3.11103835447292380131832431765, −3.02926975370978085201238999088, −2.87495001528939599360038627511, −2.80120618868688771897512637700, −2.49482508321868146884535970904, −1.96558955011284059471403993728, −1.74567591750296615400326948160, −1.74147880285704116449080851102, −1.72321776317113762491994988331, −1.67121867353695867179672534118, −1.06717187075386221947111380126, −0.992185203468132920175220695897, −0.51931753158650391366861023784, −0.46487027571410019907102627827,
0.46487027571410019907102627827, 0.51931753158650391366861023784, 0.992185203468132920175220695897, 1.06717187075386221947111380126, 1.67121867353695867179672534118, 1.72321776317113762491994988331, 1.74147880285704116449080851102, 1.74567591750296615400326948160, 1.96558955011284059471403993728, 2.49482508321868146884535970904, 2.80120618868688771897512637700, 2.87495001528939599360038627511, 3.02926975370978085201238999088, 3.11103835447292380131832431765, 3.24552442328863808924308647627, 3.58903015924606180163727973865, 3.60229699204304422184199107173, 3.79227115410962113396518390830, 4.21443030537951812119616274626, 4.32147603313826274834685607093, 4.40611057910497230908842718511, 4.46588215099134614360666220936, 4.47792951589070623289327378626, 4.69375402562755934594745210233, 4.88488207170023570045902653561
Plot not available for L-functions of degree greater than 10.
