| L(s) = 1 | + 2·4-s + 8·9-s + 2·11-s − 5·16-s − 6·19-s + 10·29-s − 2·31-s + 16·36-s + 2·41-s + 4·44-s + 28·49-s + 12·59-s + 6·61-s − 17·64-s + 14·71-s − 12·76-s + 36·79-s + 20·81-s + 40·89-s + 16·99-s + 24·101-s − 10·109-s + 20·116-s − 55·121-s − 4·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 4-s + 8/3·9-s + 0.603·11-s − 5/4·16-s − 1.37·19-s + 1.85·29-s − 0.359·31-s + 8/3·36-s + 0.312·41-s + 0.603·44-s + 4·49-s + 1.56·59-s + 0.768·61-s − 2.12·64-s + 1.66·71-s − 1.37·76-s + 4.05·79-s + 20/9·81-s + 4.23·89-s + 1.60·99-s + 2.38·101-s − 0.957·109-s + 1.85·116-s − 5·121-s − 0.359·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.342371237\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.342371237\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( ( 1 + T )^{6} \) |
| good | 2 | \( 1 - p T^{2} + 9 T^{4} - 11 T^{6} + 9 p^{2} T^{8} - p^{5} T^{10} + p^{6} T^{12} \) |
| 3 | \( 1 - 8 T^{2} + 44 T^{4} - 149 T^{6} + 44 p^{2} T^{8} - 8 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 4 p T^{2} + 352 T^{4} - 2869 T^{6} + 352 p^{2} T^{8} - 4 p^{5} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 - T + 29 T^{2} - 23 T^{3} + 29 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 + 3 T^{2} + 237 T^{4} - 961 T^{6} + 237 p^{2} T^{8} + 3 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 28 T^{2} + 812 T^{4} - 15009 T^{6} + 812 p^{2} T^{8} - 28 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 56 T^{2} + 2472 T^{4} - 63173 T^{6} + 2472 p^{2} T^{8} - 56 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 5 T + 91 T^{2} - 285 T^{3} + 91 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + T + 63 T^{2} + 115 T^{3} + 63 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 49 T^{2} + 4357 T^{4} - 133537 T^{6} + 4357 p^{2} T^{8} - 49 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - T + p T^{2} + 73 T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 7 T^{2} + 502 T^{4} - 75811 T^{6} + 502 p^{2} T^{8} - 7 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 73 T^{2} + 3537 T^{4} - 120889 T^{6} + 3537 p^{2} T^{8} - 73 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 + 37 T^{2} + 2574 T^{4} + 40561 T^{6} + 2574 p^{2} T^{8} + 37 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 6 T + 137 T^{2} - 508 T^{3} + 137 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 3 T + 173 T^{2} - 367 T^{3} + 173 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 233 T^{2} + 26437 T^{4} - 2023649 T^{6} + 26437 p^{2} T^{8} - 233 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 7 T + 212 T^{2} - 947 T^{3} + 212 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 377 T^{2} + 62917 T^{4} - 5943041 T^{6} + 62917 p^{2} T^{8} - 377 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 18 T + 254 T^{2} - 31 p T^{3} + 254 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 + 51 T^{2} - 6819 T^{4} - 697137 T^{6} - 6819 p^{2} T^{8} + 51 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 20 T + 318 T^{2} - 3435 T^{3} + 318 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 413 T^{2} + 79957 T^{4} - 9536609 T^{6} + 79957 p^{2} T^{8} - 413 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−6.05700799153281735036208714066, −5.75573087197535777027025033939, −5.71288062202192544679471992978, −5.33195578376663913744324793332, −5.08011489686288432605022771968, −5.03770085644675788851732631514, −5.00210981434479267325767906747, −4.49570962617696038617552861540, −4.39446479170640736799797195350, −4.38614780625759664393458094098, −4.12339578204318141889404929651, −4.01258524216955506088790076057, −3.65493824072221312134780137273, −3.61456565048904532477824036870, −3.49703510890838847223918196093, −3.01046236395416844157784638829, −2.63238659433193840400727091627, −2.54771212301762541711046823929, −2.30349300319789999501504063094, −2.03883293802583197803124029126, −1.95462892868371112159918293067, −1.80531395515927129298793889554, −1.13604571833398348087700464347, −0.843513459550189060569675064243, −0.77630648674142647658818347821,
0.77630648674142647658818347821, 0.843513459550189060569675064243, 1.13604571833398348087700464347, 1.80531395515927129298793889554, 1.95462892868371112159918293067, 2.03883293802583197803124029126, 2.30349300319789999501504063094, 2.54771212301762541711046823929, 2.63238659433193840400727091627, 3.01046236395416844157784638829, 3.49703510890838847223918196093, 3.61456565048904532477824036870, 3.65493824072221312134780137273, 4.01258524216955506088790076057, 4.12339578204318141889404929651, 4.38614780625759664393458094098, 4.39446479170640736799797195350, 4.49570962617696038617552861540, 5.00210981434479267325767906747, 5.03770085644675788851732631514, 5.08011489686288432605022771968, 5.33195578376663913744324793332, 5.71288062202192544679471992978, 5.75573087197535777027025033939, 6.05700799153281735036208714066
Plot not available for L-functions of degree greater than 10.
