| L(s) = 1 | + 5·9-s − 8·11-s + 6·19-s − 6·29-s − 28·31-s − 20·41-s + 17·49-s + 14·59-s + 40·61-s − 16·71-s − 28·79-s + 14·81-s − 36·89-s − 40·99-s − 8·101-s + 22·109-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 9·169-s + ⋯ |
| L(s) = 1 | + 5/3·9-s − 2.41·11-s + 1.37·19-s − 1.11·29-s − 5.02·31-s − 3.12·41-s + 17/7·49-s + 1.82·59-s + 5.12·61-s − 1.89·71-s − 3.15·79-s + 14/9·81-s − 3.81·89-s − 4.02·99-s − 0.796·101-s + 2.10·109-s + 2·121-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 + 0.0783·163-s + 0.0773·167-s + 9/13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.828520768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.828520768\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( ( 1 - T )^{6} \) |
| good | 3 | \( 1 - 5 T^{2} + 11 T^{4} - 26 T^{6} + 11 p^{2} T^{8} - 5 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 17 T^{2} + 115 T^{4} - 566 T^{6} + 115 p^{2} T^{8} - 17 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + 4 T + 13 T^{2} + 24 T^{3} + 13 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 9 T^{2} + 491 T^{4} - 2882 T^{6} + 491 p^{2} T^{8} - 9 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 45 T^{2} + 1043 T^{4} - 18878 T^{6} + 1043 p^{2} T^{8} - 45 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 25 T^{2} + 675 T^{4} - 3334 T^{6} + 675 p^{2} T^{8} - 25 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 + 3 T + 15 T^{2} + 66 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + 14 T + 125 T^{2} + 804 T^{3} + 125 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 118 T^{2} + 7819 T^{4} - 356428 T^{6} + 7819 p^{2} T^{8} - 118 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 10 T + 79 T^{2} + 348 T^{3} + 79 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 + 18 T^{2} + 4967 T^{4} + 66364 T^{6} + 4967 p^{2} T^{8} + 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 202 T^{2} + 19695 T^{4} - 1162444 T^{6} + 19695 p^{2} T^{8} - 202 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 241 T^{2} + 25979 T^{4} - 1697586 T^{6} + 25979 p^{2} T^{8} - 241 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 7 T + 185 T^{2} - 818 T^{3} + 185 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 20 T + 215 T^{2} - 1800 T^{3} + 215 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 133 T^{2} + 12139 T^{4} - 753178 T^{6} + 12139 p^{2} T^{8} - 133 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 8 T + 133 T^{2} + 624 T^{3} + 133 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 141 T^{2} + 10307 T^{4} - 530078 T^{6} + 10307 p^{2} T^{8} - 141 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 14 T + 181 T^{2} + 2228 T^{3} + 181 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 206 T^{2} + 24951 T^{4} - 2323268 T^{6} + 24951 p^{2} T^{8} - 206 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 18 T + 335 T^{2} + 3244 T^{3} + 335 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 + 2 T^{2} + 17315 T^{4} + 104772 T^{6} + 17315 p^{2} T^{8} + 2 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.30000658686166837764257869129, −4.25471154972273391520390929237, −4.16006749463299209500802865181, −4.08609229368864350604986600749, −3.61840430836568245817178760518, −3.57585510684838047568963495053, −3.55775146505649939942628148591, −3.47308920985208878434409215321, −3.34144194769584065643569725860, −3.05364215897853056653497623342, −2.90191346162540345111680790215, −2.61513484120611356373427659084, −2.56754261758885739691919033778, −2.55965775432027961233084766395, −2.17881355145485401445113914119, −1.99705044600949965226312078625, −1.84532897939226261473956125606, −1.82820213190766681649474878966, −1.58550971072670501194788095336, −1.48945296956060028923711950555, −1.17584901098911734538775764151, −0.949178259634359153352993798689, −0.54236866150373580835468853725, −0.45307212494283683265168155256, −0.16697657566335678666149884318,
0.16697657566335678666149884318, 0.45307212494283683265168155256, 0.54236866150373580835468853725, 0.949178259634359153352993798689, 1.17584901098911734538775764151, 1.48945296956060028923711950555, 1.58550971072670501194788095336, 1.82820213190766681649474878966, 1.84532897939226261473956125606, 1.99705044600949965226312078625, 2.17881355145485401445113914119, 2.55965775432027961233084766395, 2.56754261758885739691919033778, 2.61513484120611356373427659084, 2.90191346162540345111680790215, 3.05364215897853056653497623342, 3.34144194769584065643569725860, 3.47308920985208878434409215321, 3.55775146505649939942628148591, 3.57585510684838047568963495053, 3.61840430836568245817178760518, 4.08609229368864350604986600749, 4.16006749463299209500802865181, 4.25471154972273391520390929237, 4.30000658686166837764257869129
Plot not available for L-functions of degree greater than 10.
