| L(s) = 1 | − 3·4-s + 3·9-s − 10·11-s + 6·16-s + 2·19-s + 28·29-s + 12·31-s − 9·36-s + 38·41-s + 30·44-s + 22·49-s + 16·59-s + 24·61-s − 10·64-s + 16·71-s − 6·76-s − 4·79-s + 3·81-s − 2·89-s − 30·99-s + 24·101-s − 24·109-s − 84·116-s + 27·121-s − 36·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 9-s − 3.01·11-s + 3/2·16-s + 0.458·19-s + 5.19·29-s + 2.15·31-s − 3/2·36-s + 5.93·41-s + 4.52·44-s + 22/7·49-s + 2.08·59-s + 3.07·61-s − 5/4·64-s + 1.89·71-s − 0.688·76-s − 0.450·79-s + 1/3·81-s − 0.211·89-s − 3.01·99-s + 2.38·101-s − 2.29·109-s − 7.79·116-s + 2.45·121-s − 3.23·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{12} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{12} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.836119410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.836119410\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 \) |
| 37 | \( ( 1 + T^{2} )^{3} \) |
| good | 3 | \( 1 - p T^{2} + 2 p T^{4} - 35 T^{6} + 2 p^{3} T^{8} - p^{5} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 22 T^{2} + 227 T^{4} - 1672 T^{6} + 227 p^{2} T^{8} - 22 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + 5 T + 24 T^{2} + 59 T^{3} + 24 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 42 T^{2} + 891 T^{4} - 13360 T^{6} + 891 p^{2} T^{8} - 42 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 71 T^{2} + 2350 T^{4} - 48679 T^{6} + 2350 p^{2} T^{8} - 71 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 - T + 32 T^{2} - 49 T^{3} + 32 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 74 T^{2} + 3007 T^{4} - 82060 T^{6} + 3007 p^{2} T^{8} - 74 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 14 T + 123 T^{2} - 788 T^{3} + 123 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 - 6 T + 57 T^{2} - 158 T^{3} + 57 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( ( 1 - 19 T + 174 T^{2} - 1159 T^{3} + 174 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 10 T^{2} + 3143 T^{4} - 74236 T^{6} + 3143 p^{2} T^{8} - 10 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( ( 1 - p T^{2} )^{6} \) |
| 53 | \( 1 - 114 T^{2} + 8535 T^{4} - 515356 T^{6} + 8535 p^{2} T^{8} - 114 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 8 T + 117 T^{2} - 1028 T^{3} + 117 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 12 T + 165 T^{2} - 1226 T^{3} + 165 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 243 T^{2} + 30246 T^{4} - 2472547 T^{6} + 30246 p^{2} T^{8} - 243 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 8 T + 159 T^{2} - 758 T^{3} + 159 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 331 T^{2} + 49886 T^{4} - 4533511 T^{6} + 49886 p^{2} T^{8} - 331 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 2 T + 137 T^{2} + 404 T^{3} + 137 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 287 T^{2} + 36514 T^{4} - 3246643 T^{6} + 36514 p^{2} T^{8} - 287 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + T + 78 T^{2} + 31 T^{3} + 78 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 378 T^{2} + 62223 T^{4} - 6783340 T^{6} + 62223 p^{2} T^{8} - 378 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.64601734439768776949004138676, −4.57831908574141335514919526129, −4.50643348685175503499766205131, −4.48920825732716379166754598802, −4.31934063094934198991826669281, −4.25024716455997960448699638693, −4.08100744676141748616844009783, −3.62740916903624920960785403376, −3.61701446792268756661767197073, −3.46672394732730356669730035858, −3.34621901241882838187594418337, −2.85293382863246096218326658232, −2.80021561467519197821358914369, −2.73951498632441218551425402704, −2.53446196530576023638673568231, −2.48979122939384225606567555788, −2.29858666298445224730728806971, −2.10435675571612180496112392587, −1.99738701482508683225535501905, −1.15955527483759581664365949180, −1.10989498125410361064204116807, −0.956512961306164535443886706869, −0.871123833929993193664677498304, −0.59134992460358639107036606780, −0.50295495698832067412074754219,
0.50295495698832067412074754219, 0.59134992460358639107036606780, 0.871123833929993193664677498304, 0.956512961306164535443886706869, 1.10989498125410361064204116807, 1.15955527483759581664365949180, 1.99738701482508683225535501905, 2.10435675571612180496112392587, 2.29858666298445224730728806971, 2.48979122939384225606567555788, 2.53446196530576023638673568231, 2.73951498632441218551425402704, 2.80021561467519197821358914369, 2.85293382863246096218326658232, 3.34621901241882838187594418337, 3.46672394732730356669730035858, 3.61701446792268756661767197073, 3.62740916903624920960785403376, 4.08100744676141748616844009783, 4.25024716455997960448699638693, 4.31934063094934198991826669281, 4.48920825732716379166754598802, 4.50643348685175503499766205131, 4.57831908574141335514919526129, 4.64601734439768776949004138676
Plot not available for L-functions of degree greater than 10.
