| L(s) = 1 | + 4-s − 6·5-s − 4·16-s − 6·20-s + 4·23-s + 21·25-s + 18·31-s + 10·37-s + 13·49-s + 16·53-s + 20·59-s − 5·64-s + 10·67-s + 4·71-s + 24·80-s − 48·89-s + 4·92-s + 2·97-s + 21·100-s + 14·103-s − 20·113-s − 24·115-s + 18·124-s − 56·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 2.68·5-s − 16-s − 1.34·20-s + 0.834·23-s + 21/5·25-s + 3.23·31-s + 1.64·37-s + 13/7·49-s + 2.19·53-s + 2.60·59-s − 5/8·64-s + 1.22·67-s + 0.474·71-s + 2.68·80-s − 5.08·89-s + 0.417·92-s + 0.203·97-s + 2.09·100-s + 1.37·103-s − 1.88·113-s − 2.23·115-s + 1.61·124-s − 5.00·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.287201036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.287201036\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 + T )^{6} \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - T^{2} + 5 T^{4} - p^{2} T^{6} + 5 p^{2} T^{8} - p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 13 T^{2} + 183 T^{4} - 1286 T^{6} + 183 p^{2} T^{8} - 13 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( 1 + 2 p T^{2} + 615 T^{4} + 8524 T^{6} + 615 p^{2} T^{8} + 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 + 20 T^{2} + 824 T^{4} + 9854 T^{6} + 824 p^{2} T^{8} + 20 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( 1 - T^{2} + 123 T^{4} + 94 p T^{6} + 123 p^{2} T^{8} - p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 - 2 T + 40 T^{2} - 26 T^{3} + 40 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( 1 + 122 T^{2} + 7367 T^{4} + 267788 T^{6} + 7367 p^{2} T^{8} + 122 p^{4} T^{10} + p^{6} T^{12} \) |
| 31 | \( ( 1 - 9 T + 48 T^{2} - 7 p T^{3} + 48 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 - 5 T + 89 T^{2} - 278 T^{3} + 89 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{3} \) |
| 43 | \( 1 + 66 T^{2} + 1623 T^{4} + 38972 T^{6} + 1623 p^{2} T^{8} + 66 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( ( 1 + 72 T^{2} - 216 T^{3} + 72 p T^{4} + p^{3} T^{6} )^{2} \) |
| 53 | \( ( 1 - 8 T + 160 T^{2} - 794 T^{3} + 160 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( ( 1 - 10 T + 61 T^{2} - 268 T^{3} + 61 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( 1 + 125 T^{2} + 12702 T^{4} + 764233 T^{6} + 12702 p^{2} T^{8} + 125 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 - 5 T + 179 T^{2} - 578 T^{3} + 179 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 71 | \( ( 1 - 2 T + 133 T^{2} - 380 T^{3} + 133 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 + 291 T^{2} + 42627 T^{4} + 3837890 T^{6} + 42627 p^{2} T^{8} + 291 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( 1 + 65 T^{2} + 6462 T^{4} + 16357 T^{6} + 6462 p^{2} T^{8} + 65 p^{4} T^{10} + p^{6} T^{12} \) |
| 83 | \( 1 + 314 T^{2} + 50711 T^{4} + 5124716 T^{6} + 50711 p^{2} T^{8} + 314 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 24 T + 387 T^{2} + 4056 T^{3} + 387 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( ( 1 - T + 65 T^{2} + 650 T^{3} + 65 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
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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
−4.20642574897329589368631919430, −3.98873137780149301001122475493, −3.86570379471397627202342331609, −3.78858501042166275181201546415, −3.75086133741116756570626327906, −3.55754880920883179771567495524, −3.40664657475313263642948324569, −3.14403424512550622183319930422, −2.95712931477861764267466825946, −2.93774317896234412961343989215, −2.73545824988599632353831661088, −2.59448877823452399610689354528, −2.52516090515391900787392414783, −2.33364538634480021987012428947, −2.28472193701137755478961608090, −2.22526224566019542574758983700, −1.70567574809066292596865643005, −1.57045539620103808021048764250, −1.25540207507925440030086539357, −1.24535422135233317472652416422, −1.02009185806029167993997176030, −0.70420027354665521076021446185, −0.68989211239839315234940367968, −0.36743521983165110447351417232, −0.35118143276415731066267972757,
0.35118143276415731066267972757, 0.36743521983165110447351417232, 0.68989211239839315234940367968, 0.70420027354665521076021446185, 1.02009185806029167993997176030, 1.24535422135233317472652416422, 1.25540207507925440030086539357, 1.57045539620103808021048764250, 1.70567574809066292596865643005, 2.22526224566019542574758983700, 2.28472193701137755478961608090, 2.33364538634480021987012428947, 2.52516090515391900787392414783, 2.59448877823452399610689354528, 2.73545824988599632353831661088, 2.93774317896234412961343989215, 2.95712931477861764267466825946, 3.14403424512550622183319930422, 3.40664657475313263642948324569, 3.55754880920883179771567495524, 3.75086133741116756570626327906, 3.78858501042166275181201546415, 3.86570379471397627202342331609, 3.98873137780149301001122475493, 4.20642574897329589368631919430
Plot not available for L-functions of degree greater than 10.
