| L(s) = 1 | − 2·2-s − 3·4-s + 8·8-s + 9-s + 2·13-s + 2·16-s − 2·17-s − 2·18-s + 12·19-s − 4·26-s − 8·32-s + 4·34-s − 3·36-s − 24·38-s − 8·43-s + 40·47-s + 23·49-s − 6·52-s − 10·53-s − 4·59-s − 6·64-s + 24·67-s + 6·68-s + 8·72-s − 36·76-s − 3·81-s − 20·83-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 3/2·4-s + 2.82·8-s + 1/3·9-s + 0.554·13-s + 1/2·16-s − 0.485·17-s − 0.471·18-s + 2.75·19-s − 0.784·26-s − 1.41·32-s + 0.685·34-s − 1/2·36-s − 3.89·38-s − 1.21·43-s + 5.83·47-s + 23/7·49-s − 0.832·52-s − 1.37·53-s − 0.520·59-s − 3/4·64-s + 2.93·67-s + 0.727·68-s + 0.942·72-s − 4.12·76-s − 1/3·81-s − 2.19·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4954030794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4954030794\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 + 2 T + 15 T^{2} - 36 T^{3} + 15 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 2 | \( ( 1 + T + 3 T^{2} + 3 T^{3} + 3 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 3 | \( 1 - T^{2} + 4 T^{4} + 7 T^{6} + 4 p^{2} T^{8} - p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 23 T^{2} + 286 T^{4} - 2369 T^{6} + 286 p^{2} T^{8} - 23 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 + 203 T^{4} - 428 T^{6} + 203 p^{2} T^{8} + p^{6} T^{12} \) |
| 13 | \( ( 1 - T + 2 p T^{2} - 3 T^{3} + 2 p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( ( 1 - 6 T + 41 T^{2} - 128 T^{3} + 41 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 84 T^{2} + 3203 T^{4} - 82532 T^{6} + 3203 p^{2} T^{8} - 84 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( 1 - 2 p T^{2} + 1791 T^{4} - 54028 T^{6} + 1791 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 31 | \( 1 - 97 T^{2} + 4756 T^{4} - 166345 T^{6} + 4756 p^{2} T^{8} - 97 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( 1 - 78 T^{2} + 3311 T^{4} - 131684 T^{6} + 3311 p^{2} T^{8} - 78 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 146 T^{2} + 11847 T^{4} - 591644 T^{6} + 11847 p^{2} T^{8} - 146 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 + 4 T + 41 T^{2} - 108 T^{3} + 41 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( ( 1 - 20 T + 261 T^{2} - 2088 T^{3} + 261 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( ( 1 + 5 T + 90 T^{2} + 573 T^{3} + 90 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( ( 1 + 2 T + 133 T^{2} + 216 T^{3} + 133 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( 1 - 138 T^{2} + 13191 T^{4} - 929068 T^{6} + 13191 p^{2} T^{8} - 138 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 - 12 T + 233 T^{2} - 1592 T^{3} + 233 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 71 | \( 1 - 335 T^{2} + 50838 T^{4} - 4555673 T^{6} + 50838 p^{2} T^{8} - 335 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( 1 - 406 T^{2} + 70919 T^{4} - 6803988 T^{6} + 70919 p^{2} T^{8} - 406 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( 1 - 339 T^{2} + 51446 T^{4} - 4883081 T^{6} + 51446 p^{2} T^{8} - 339 p^{4} T^{10} + p^{6} T^{12} \) |
| 83 | \( ( 1 + 10 T + 149 T^{2} + 1844 T^{3} + 149 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( ( 1 + 4 T + 263 T^{2} + 692 T^{3} + 263 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 250 T^{2} + 48079 T^{4} - 5208940 T^{6} + 48079 p^{2} T^{8} - 250 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−6.10133655695673974867401004248, −6.08538601996168966526959932757, −5.52188874781743861318658386003, −5.47778936992801384962393316123, −5.20037599032290634139940591865, −5.19398553718025902749997857048, −5.12812709971993763825023292629, −4.81725843405412052491277975555, −4.60195739282756116414221586722, −4.29055831641941957705637513403, −4.11501371302613265494326630782, −4.05394989446714981212819123688, −3.79095830955073828962750212025, −3.75554655579716091157645102437, −3.46657527955908020102966059679, −3.23209841217578580637727543108, −2.72034273614031902017120468294, −2.57184021284075513317618326857, −2.52046630249875102085205311784, −2.31235295365373587138834109000, −1.60232175629293469886525525029, −1.36178026124117959750039323678, −0.925795146808570749538986615168, −0.912105174608685865190191561326, −0.36731966096398744806428901814,
0.36731966096398744806428901814, 0.912105174608685865190191561326, 0.925795146808570749538986615168, 1.36178026124117959750039323678, 1.60232175629293469886525525029, 2.31235295365373587138834109000, 2.52046630249875102085205311784, 2.57184021284075513317618326857, 2.72034273614031902017120468294, 3.23209841217578580637727543108, 3.46657527955908020102966059679, 3.75554655579716091157645102437, 3.79095830955073828962750212025, 4.05394989446714981212819123688, 4.11501371302613265494326630782, 4.29055831641941957705637513403, 4.60195739282756116414221586722, 4.81725843405412052491277975555, 5.12812709971993763825023292629, 5.19398553718025902749997857048, 5.20037599032290634139940591865, 5.47778936992801384962393316123, 5.52188874781743861318658386003, 6.08538601996168966526959932757, 6.10133655695673974867401004248
Plot not available for L-functions of degree greater than 10.
