| L(s) = 1 | − 4·9-s + 18·11-s + 6·19-s − 4·29-s + 8·31-s + 4·41-s − 15·49-s + 28·59-s + 30·61-s + 44·71-s + 24·79-s + 13·81-s + 8·89-s − 72·99-s + 8·101-s + 149·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s − 24·171-s + ⋯ |
| L(s) = 1 | − 4/3·9-s + 5.42·11-s + 1.37·19-s − 0.742·29-s + 1.43·31-s + 0.624·41-s − 2.14·49-s + 3.64·59-s + 3.84·61-s + 5.22·71-s + 2.70·79-s + 13/9·81-s + 0.847·89-s − 7.23·99-s + 0.796·101-s + 13.5·121-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 − 4·169-s − 1.83·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(18.90920861\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.90920861\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T^{2} + p T^{4} - 16 T^{6} + p^{3} T^{8} + 4 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 + 15 T^{2} + 83 T^{4} + 314 T^{6} + 83 p^{2} T^{8} + 15 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 - 9 T + 47 T^{2} - 170 T^{3} + 47 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 + 4 p T^{2} + 1291 T^{4} + 20320 T^{6} + 1291 p^{2} T^{8} + 4 p^{5} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 + 3 p T^{2} + 1595 T^{4} + 31442 T^{6} + 1595 p^{2} T^{8} + 3 p^{5} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 + 94 T^{2} + 4335 T^{4} + 124036 T^{6} + 4335 p^{2} T^{8} + 94 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 + 2 T + 3 T^{2} + 204 T^{3} + 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 - 4 T + 13 T^{2} + 8 T^{3} + 13 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 + 208 T^{2} + 18499 T^{4} + 900712 T^{6} + 18499 p^{2} T^{8} + 208 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 2 T + 7 T^{2} + 324 T^{3} + 7 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 + 31 T^{2} + 1211 T^{4} + 145386 T^{6} + 1211 p^{2} T^{8} + 31 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 + 39 T^{2} + 1475 T^{4} + 55658 T^{6} + 1475 p^{2} T^{8} + 39 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 + 204 T^{2} + 21467 T^{4} + 1404080 T^{6} + 21467 p^{2} T^{8} + 204 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 14 T + 193 T^{2} - 1524 T^{3} + 193 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 15 T + 245 T^{2} - 1874 T^{3} + 245 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 + 24 T^{2} + 1691 T^{4} + 389864 T^{6} + 1691 p^{2} T^{8} + 24 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 22 T + 325 T^{2} - 3156 T^{3} + 325 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 + 187 T^{2} + 22987 T^{4} + 2022754 T^{6} + 22987 p^{2} T^{8} + 187 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 4 T + p T^{2} )^{6} \) |
| 83 | \( 1 + 454 T^{2} + 89175 T^{4} + 9691156 T^{6} + 89175 p^{2} T^{8} + 454 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 4 T + 223 T^{2} - 648 T^{3} + 223 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 + 468 T^{2} + 100403 T^{4} + 12473600 T^{6} + 100403 p^{2} T^{8} + 468 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.89441625447526179531833013217, −4.55157960116309887987025672363, −4.47230674440475733427138606442, −4.24804863344946944319133258619, −4.14016570498397586212028156149, −4.02293047528074274878704118832, −3.76938485448122477379246609713, −3.69879517333681734992166708263, −3.50519090073751939483112350586, −3.47170033987673154571460218050, −3.46653409816443336706769462798, −3.18070214879256188236166565034, −3.13215551145543620606969761439, −2.43139520037091939997664557467, −2.41491651276156604287337794226, −2.37422277111233868413714740535, −2.24622620408940308031995104146, −2.06490522029375114366641500384, −1.70233081871925013059287140277, −1.36604169372458639386397768431, −1.35757910354703740290716756116, −0.983205023654076412542980964984, −0.880951095602506269429996086031, −0.68904144651399685866559148777, −0.53756157683053079461003563806,
0.53756157683053079461003563806, 0.68904144651399685866559148777, 0.880951095602506269429996086031, 0.983205023654076412542980964984, 1.35757910354703740290716756116, 1.36604169372458639386397768431, 1.70233081871925013059287140277, 2.06490522029375114366641500384, 2.24622620408940308031995104146, 2.37422277111233868413714740535, 2.41491651276156604287337794226, 2.43139520037091939997664557467, 3.13215551145543620606969761439, 3.18070214879256188236166565034, 3.46653409816443336706769462798, 3.47170033987673154571460218050, 3.50519090073751939483112350586, 3.69879517333681734992166708263, 3.76938485448122477379246609713, 4.02293047528074274878704118832, 4.14016570498397586212028156149, 4.24804863344946944319133258619, 4.47230674440475733427138606442, 4.55157960116309887987025672363, 4.89441625447526179531833013217
Plot not available for L-functions of degree greater than 10.
