| L(s) = 1 | − 3·4-s − 4·5-s − 3·9-s + 16·11-s + 6·16-s + 4·19-s + 12·20-s + 4·25-s + 4·29-s − 6·31-s + 9·36-s + 12·41-s − 48·44-s + 12·45-s + 30·49-s − 64·55-s + 4·59-s + 28·61-s − 10·64-s − 16·71-s − 12·76-s + 12·79-s − 24·80-s + 6·81-s + 4·89-s − 16·95-s − 48·99-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 1.78·5-s − 9-s + 4.82·11-s + 3/2·16-s + 0.917·19-s + 2.68·20-s + 4/5·25-s + 0.742·29-s − 1.07·31-s + 3/2·36-s + 1.87·41-s − 7.23·44-s + 1.78·45-s + 30/7·49-s − 8.62·55-s + 0.520·59-s + 3.58·61-s − 5/4·64-s − 1.89·71-s − 1.37·76-s + 1.35·79-s − 2.68·80-s + 2/3·81-s + 0.423·89-s − 1.64·95-s − 4.82·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.992161861\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.992161861\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 3 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 + 4 T + 12 T^{2} + 36 T^{3} + 12 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | \( ( 1 + T )^{6} \) |
| good | 7 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{3} \) |
| 11 | \( ( 1 - 8 T + 46 T^{2} - 168 T^{3} + 46 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 28 T^{2} + 280 T^{4} - 2014 T^{6} + 280 p^{2} T^{8} - 28 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 28 T^{2} + 512 T^{4} - 11034 T^{6} + 512 p^{2} T^{8} - 28 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 - 2 T + 22 T^{2} - 80 T^{3} + 22 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 70 T^{2} + 2639 T^{4} - 71412 T^{6} + 2639 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 2 T + 55 T^{2} - 132 T^{3} + 55 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 58 T^{2} + 1895 T^{4} - 66348 T^{6} + 1895 p^{2} T^{8} - 58 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 2 T + p T^{2} )^{6} \) |
| 43 | \( 1 - 106 T^{2} + 7367 T^{4} - 400332 T^{6} + 7367 p^{2} T^{8} - 106 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 208 T^{2} + 20432 T^{4} - 1209294 T^{6} + 20432 p^{2} T^{8} - 208 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 130 T^{2} + 4055 T^{4} + 2820 T^{6} + 4055 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 2 T + 101 T^{2} + 44 T^{3} + 101 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 14 T + 92 T^{2} - 336 T^{3} + 92 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 116 T^{2} + 11776 T^{4} - 712958 T^{6} + 11776 p^{2} T^{8} - 116 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 8 T + 134 T^{2} + 1156 T^{3} + 134 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 170 T^{2} + 20671 T^{4} - 1830476 T^{6} + 20671 p^{2} T^{8} - 170 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 6 T + 230 T^{2} - 920 T^{3} + 230 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 128 T^{2} + 13008 T^{4} - 1479782 T^{6} + 13008 p^{2} T^{8} - 128 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 2 T + 123 T^{2} - 36 T^{3} + 123 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 432 T^{2} + 89120 T^{4} - 10929554 T^{6} + 89120 p^{2} T^{8} - 432 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−5.28579331807154264471005186648, −5.17898751134358155285796112113, −5.05774721293566686167443644511, −4.82320216324444104996066113937, −4.50416628739632516664040818881, −4.39630658267190707424944343763, −4.15923503549313541264775914404, −4.02776596307716053345255355874, −4.01845292749549910138937693410, −3.77988037861326749413454058666, −3.76122966367400274574582673828, −3.72840688322061401172338255051, −3.66575857188705674443097216072, −2.99862139772311652847097024544, −2.94188179586906768208890435097, −2.82763970041976481777680247682, −2.73154872617466493573894720637, −2.09185019993051032073452075394, −2.04695849668419524796122432955, −1.63117750483127575910222366453, −1.57738892623066139293423674829, −1.01461484730051513890731435765, −0.78776609613215060268633625533, −0.75028415678710255794022844439, −0.53985296122424655334859698180,
0.53985296122424655334859698180, 0.75028415678710255794022844439, 0.78776609613215060268633625533, 1.01461484730051513890731435765, 1.57738892623066139293423674829, 1.63117750483127575910222366453, 2.04695849668419524796122432955, 2.09185019993051032073452075394, 2.73154872617466493573894720637, 2.82763970041976481777680247682, 2.94188179586906768208890435097, 2.99862139772311652847097024544, 3.66575857188705674443097216072, 3.72840688322061401172338255051, 3.76122966367400274574582673828, 3.77988037861326749413454058666, 4.01845292749549910138937693410, 4.02776596307716053345255355874, 4.15923503549313541264775914404, 4.39630658267190707424944343763, 4.50416628739632516664040818881, 4.82320216324444104996066113937, 5.05774721293566686167443644511, 5.17898751134358155285796112113, 5.28579331807154264471005186648
Plot not available for L-functions of degree greater than 10.
