| L(s) = 1 | + 4·7-s + 12·17-s + 8·23-s + 12·31-s + 20·41-s − 8·47-s + 2·49-s − 8·71-s + 36·73-s − 36·79-s + 28·89-s − 36·97-s − 4·103-s + 52·113-s + 48·119-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 32·161-s + 163-s + 167-s + 22·169-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 2.91·17-s + 1.66·23-s + 2.15·31-s + 3.12·41-s − 1.16·47-s + 2/7·49-s − 0.949·71-s + 4.21·73-s − 4.05·79-s + 2.96·89-s − 3.65·97-s − 0.394·103-s + 4.89·113-s + 4.40·119-s + 2/11·121-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 + 2.52·161-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.04125419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.04125419\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 - 2 T + 5 T^{2} - 12 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 11 | \( 1 - 2 T^{2} + 87 T^{4} + 4 T^{6} + 87 p^{2} T^{8} - 2 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( 1 - 22 T^{2} + 407 T^{4} - 7284 T^{6} + 407 p^{2} T^{8} - 22 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 - 6 T + 35 T^{2} - 172 T^{3} + 35 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( 1 - 74 T^{2} + 2647 T^{4} - 60620 T^{6} + 2647 p^{2} T^{8} - 74 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 - 4 T + 57 T^{2} - 168 T^{3} + 57 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{3} \) |
| 31 | \( ( 1 - 6 T + 77 T^{2} - 308 T^{3} + 77 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 86 T^{2} + 6055 T^{4} - 248372 T^{6} + 6055 p^{2} T^{8} - 86 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 10 T + 87 T^{2} - 588 T^{3} + 87 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 114 T^{2} + 8087 T^{4} - 416540 T^{6} + 8087 p^{2} T^{8} - 114 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( ( 1 + 4 T + 49 T^{2} - 120 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{3} \) |
| 59 | \( 1 - 274 T^{2} + 33911 T^{4} - 2503644 T^{6} + 33911 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 - 110 T^{2} + 10759 T^{4} - 685796 T^{6} + 10759 p^{2} T^{8} - 110 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{3} \) |
| 71 | \( ( 1 + 4 T + 101 T^{2} + 632 T^{3} + 101 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 - 6 T + p T^{2} )^{6} \) |
| 79 | \( ( 1 + 18 T + 317 T^{2} + 2908 T^{3} + 317 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 274 T^{2} + 37415 T^{4} - 3513756 T^{6} + 37415 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 14 T + 263 T^{2} - 2308 T^{3} + 263 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( ( 1 + 18 T + 287 T^{2} + 3164 T^{3} + 287 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
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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
−3.95429138477709466938564095500, −3.90611107150007893189547401643, −3.87364920090578492588822979184, −3.64236859473546134590687693704, −3.42042572764493985305555754546, −3.30072697707591921596681194477, −3.21567407608056755690778259755, −3.18193335080929511345923529794, −2.92879981315349072503112350433, −2.66884953527637305723649818248, −2.59265841191867027365937453863, −2.53616866702306468831509828492, −2.49457865670244400671384909813, −2.28280176566999872321957581879, −2.00411685406739795977869107082, −1.92243190110837109144139987108, −1.59882775035723818790569833803, −1.31758176966735746381519597889, −1.30627936697431948076159889722, −1.25937066239989257878753328726, −1.11394030276598942753880896861, −1.01314077577004226059198036795, −0.63448110317034944870442344453, −0.52284931552214280159840540939, −0.21058909568908361275071462275,
0.21058909568908361275071462275, 0.52284931552214280159840540939, 0.63448110317034944870442344453, 1.01314077577004226059198036795, 1.11394030276598942753880896861, 1.25937066239989257878753328726, 1.30627936697431948076159889722, 1.31758176966735746381519597889, 1.59882775035723818790569833803, 1.92243190110837109144139987108, 2.00411685406739795977869107082, 2.28280176566999872321957581879, 2.49457865670244400671384909813, 2.53616866702306468831509828492, 2.59265841191867027365937453863, 2.66884953527637305723649818248, 2.92879981315349072503112350433, 3.18193335080929511345923529794, 3.21567407608056755690778259755, 3.30072697707591921596681194477, 3.42042572764493985305555754546, 3.64236859473546134590687693704, 3.87364920090578492588822979184, 3.90611107150007893189547401643, 3.95429138477709466938564095500
Plot not available for L-functions of degree greater than 10.
