| L(s) = 1 | + 2·3-s − 6·5-s + 3·9-s − 12·15-s − 4·23-s + 21·25-s + 10·27-s − 28·31-s + 32·37-s − 18·45-s + 6·47-s − 13·49-s − 4·53-s − 12·59-s + 26·67-s − 8·69-s + 12·71-s + 42·75-s + 23·81-s + 10·89-s − 56·93-s + 44·97-s − 52·103-s + 64·111-s + 28·113-s + 24·115-s − 56·125-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 2.68·5-s + 9-s − 3.09·15-s − 0.834·23-s + 21/5·25-s + 1.92·27-s − 5.02·31-s + 5.26·37-s − 2.68·45-s + 0.875·47-s − 1.85·49-s − 0.549·53-s − 1.56·59-s + 3.17·67-s − 0.963·69-s + 1.42·71-s + 4.84·75-s + 23/9·81-s + 1.05·89-s − 5.80·93-s + 4.46·97-s − 5.12·103-s + 6.07·111-s + 2.63·113-s + 2.23·115-s − 5.00·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.796660503\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.796660503\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + T )^{6} \) |
| 11 | \( 1 \) |
| good | 3 | \( ( 1 - T - p T^{3} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 7 | \( 1 + 13 T^{2} + 158 T^{4} + 1117 T^{6} + 158 p^{2} T^{8} + 13 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( 1 + 10 T^{2} + 503 T^{4} + 3292 T^{6} + 503 p^{2} T^{8} + 10 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 + 22 T^{2} + 495 T^{4} + 7028 T^{6} + 495 p^{2} T^{8} + 22 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( 1 + 10 T^{2} + 1079 T^{4} + 7132 T^{6} + 1079 p^{2} T^{8} + 10 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 + 2 T + 25 T^{2} + 104 T^{3} + 25 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( 1 + 94 T^{2} + 4935 T^{4} + 169988 T^{6} + 4935 p^{2} T^{8} + 94 p^{4} T^{10} + p^{6} T^{12} \) |
| 31 | \( ( 1 + 14 T + 125 T^{2} + 848 T^{3} + 125 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 - 16 T + 159 T^{2} - 1136 T^{3} + 159 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 + 133 T^{2} + 6846 T^{4} + 252041 T^{6} + 6846 p^{2} T^{8} + 133 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( 1 + 157 T^{2} + 12974 T^{4} + 688117 T^{6} + 12974 p^{2} T^{8} + 157 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( ( 1 - 3 T + 60 T^{2} - 201 T^{3} + 60 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( ( 1 + 2 T + 115 T^{2} + 224 T^{3} + 115 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( ( 1 + 6 T + 93 T^{2} + 168 T^{3} + 93 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( 1 + 277 T^{2} + 36422 T^{4} + 2817289 T^{6} + 36422 p^{2} T^{8} + 277 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 - 13 T + 140 T^{2} - 847 T^{3} + 140 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 71 | \( ( 1 - 6 T + 129 T^{2} - 312 T^{3} + 129 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{3} \) |
| 79 | \( 1 + 22 T^{2} - 721 T^{4} + 597364 T^{6} - 721 p^{2} T^{8} + 22 p^{4} T^{10} + p^{6} T^{12} \) |
| 83 | \( 1 + 154 T^{2} + 19095 T^{4} + 2118524 T^{6} + 19095 p^{2} T^{8} + 154 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 5 T + 238 T^{2} - 833 T^{3} + 238 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( ( 1 - 22 T + 407 T^{2} - 4288 T^{3} + 407 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
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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
−3.87099585918686449259819795504, −3.84849562144555085602904400766, −3.50645014439262634346272842627, −3.49002918531380817584790200137, −3.35857587814011342807033188200, −3.32652862716918976347383333410, −3.31604781861549141416440760950, −3.01258550975391915385379459816, −2.76863931349823043566762662249, −2.74294197949376265760062130722, −2.56705231114062286865624056804, −2.48423734475476580047533570694, −2.31219607191845010274930464242, −2.01250659434059008165543651615, −1.98805207633491434456815011333, −1.97293154074098612475722838992, −1.59944006232141554947189814307, −1.52719097002605503647084665591, −1.38705860958705886151318579572, −1.02460905576222631865649010416, −0.898459652475036108815849831623, −0.65517986914311191993616089569, −0.55304061499723680554186985119, −0.45201600494989195267218535646, −0.21948547685300610240649924108,
0.21948547685300610240649924108, 0.45201600494989195267218535646, 0.55304061499723680554186985119, 0.65517986914311191993616089569, 0.898459652475036108815849831623, 1.02460905576222631865649010416, 1.38705860958705886151318579572, 1.52719097002605503647084665591, 1.59944006232141554947189814307, 1.97293154074098612475722838992, 1.98805207633491434456815011333, 2.01250659434059008165543651615, 2.31219607191845010274930464242, 2.48423734475476580047533570694, 2.56705231114062286865624056804, 2.74294197949376265760062130722, 2.76863931349823043566762662249, 3.01258550975391915385379459816, 3.31604781861549141416440760950, 3.32652862716918976347383333410, 3.35857587814011342807033188200, 3.49002918531380817584790200137, 3.50645014439262634346272842627, 3.84849562144555085602904400766, 3.87099585918686449259819795504
Plot not available for L-functions of degree greater than 10.
