| L(s) = 1 | − 3·4-s − 12·11-s + 6·16-s − 4·19-s + 25-s + 20·29-s − 12·31-s − 28·41-s + 36·44-s + 26·49-s + 28·59-s − 4·61-s − 10·64-s − 8·71-s + 12·76-s + 16·79-s + 44·89-s − 3·100-s − 44·101-s + 12·109-s − 60·116-s + 18·121-s + 36·124-s − 16·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 3.61·11-s + 3/2·16-s − 0.917·19-s + 1/5·25-s + 3.71·29-s − 2.15·31-s − 4.37·41-s + 5.42·44-s + 26/7·49-s + 3.64·59-s − 0.512·61-s − 5/4·64-s − 0.949·71-s + 1.37·76-s + 1.80·79-s + 4.66·89-s − 0.299·100-s − 4.37·101-s + 1.14·109-s − 5.57·116-s + 1.63·121-s + 3.23·124-s − 1.43·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.676372951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.676372951\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T^{2} + 16 T^{3} - p T^{4} + p^{3} T^{6} \) |
| 13 | \( ( 1 + T^{2} )^{3} \) |
| good | 7 | \( 1 - 26 T^{2} + 319 T^{4} - 2588 T^{6} + 319 p^{2} T^{8} - 26 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + 2 T + p T^{2} )^{6} \) |
| 17 | \( 1 - 30 T^{2} + 1151 T^{4} - 18164 T^{6} + 1151 p^{2} T^{8} - 30 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 + 2 T + 49 T^{2} + 80 T^{3} + 49 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 58 T^{2} + 2175 T^{4} - 57244 T^{6} + 2175 p^{2} T^{8} - 58 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 10 T + 43 T^{2} - 192 T^{3} + 43 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + 6 T + 89 T^{2} + 332 T^{3} + 89 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 82 T^{2} + 2935 T^{4} - 75484 T^{6} + 2935 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 14 T + 175 T^{2} + 1188 T^{3} + 175 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 18 T^{2} + 855 T^{4} - 165724 T^{6} + 855 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 234 T^{2} + 24623 T^{4} - 1487372 T^{6} + 24623 p^{2} T^{8} - 234 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 46 T^{2} + 2967 T^{4} - 107812 T^{6} + 2967 p^{2} T^{8} - 46 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 14 T + 205 T^{2} - 1500 T^{3} + 205 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 2 T + 51 T^{2} - 436 T^{3} + 51 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 262 T^{2} + 32423 T^{4} - 2593908 T^{6} + 32423 p^{2} T^{8} - 262 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 4 T + 133 T^{2} + 168 T^{3} + 133 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 294 T^{2} + 40479 T^{4} - 3542852 T^{6} + 40479 p^{2} T^{8} - 294 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 8 T - 3 T^{2} + 448 T^{3} - 3 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 370 T^{2} + 63335 T^{4} - 6552924 T^{6} + 63335 p^{2} T^{8} - 370 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 22 T + 391 T^{2} - 4116 T^{3} + 391 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 310 T^{2} + 58351 T^{4} - 6728164 T^{6} + 58351 p^{2} T^{8} - 310 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−5.15516370501241418437467989610, −5.06835332678259619329547873917, −5.06754758766786677439044323983, −4.79845409521555075451478553764, −4.36635239856736546953965435663, −4.35911795972726422442677900436, −4.33768758611873188842958745859, −4.10164632192913681471296729668, −3.73645098731298056771776211702, −3.72911011995714835084532732479, −3.58642940865768011749628786507, −3.43323423641007992587080617685, −2.98523700247761674794036627095, −2.81781449182612074086500396360, −2.79436357547756627913271611364, −2.63133787634348169894773941563, −2.60245615309758543206264223921, −2.14131810951456483567287302779, −1.87382396568432829483087226301, −1.81145155570143186534866480980, −1.61846595725382923159618177305, −1.05328662207236730763614273886, −0.59569675783565761835236191891, −0.54965333296788914856874267360, −0.38750489630033830241152374892,
0.38750489630033830241152374892, 0.54965333296788914856874267360, 0.59569675783565761835236191891, 1.05328662207236730763614273886, 1.61846595725382923159618177305, 1.81145155570143186534866480980, 1.87382396568432829483087226301, 2.14131810951456483567287302779, 2.60245615309758543206264223921, 2.63133787634348169894773941563, 2.79436357547756627913271611364, 2.81781449182612074086500396360, 2.98523700247761674794036627095, 3.43323423641007992587080617685, 3.58642940865768011749628786507, 3.72911011995714835084532732479, 3.73645098731298056771776211702, 4.10164632192913681471296729668, 4.33768758611873188842958745859, 4.35911795972726422442677900436, 4.36635239856736546953965435663, 4.79845409521555075451478553764, 5.06754758766786677439044323983, 5.06835332678259619329547873917, 5.15516370501241418437467989610
Plot not available for L-functions of degree greater than 10.
