| L(s) = 1 | − 6·3-s + 21·9-s − 8·13-s + 25-s − 56·27-s + 16·31-s − 16·37-s + 48·39-s − 4·41-s + 18·49-s + 24·53-s − 16·71-s − 6·75-s − 16·79-s + 126·81-s + 16·83-s − 20·89-s − 96·93-s + 24·107-s + 96·111-s − 168·117-s + 34·121-s + 24·123-s + 8·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 3.46·3-s + 7·9-s − 2.21·13-s + 1/5·25-s − 10.7·27-s + 2.87·31-s − 2.63·37-s + 7.68·39-s − 0.624·41-s + 18/7·49-s + 3.29·53-s − 1.89·71-s − 0.692·75-s − 1.80·79-s + 14·81-s + 1.75·83-s − 2.11·89-s − 9.95·93-s + 2.32·107-s + 9.11·111-s − 15.5·117-s + 3.09·121-s + 2.16·123-s + 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{6} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{6} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4498524053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4498524053\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T )^{6} \) |
| 5 | \( 1 - T^{2} - 8 T^{3} - p T^{4} + p^{3} T^{6} \) |
| good | 7 | \( 1 - 18 T^{2} + 191 T^{4} - 1532 T^{6} + 191 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 34 T^{2} + 503 T^{4} - 5436 T^{6} + 503 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( ( 1 + 4 T + 23 T^{2} + 48 T^{3} + 23 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 17 | \( 1 - 66 T^{2} + 2255 T^{4} - 47324 T^{6} + 2255 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( 1 - 54 T^{2} + 1367 T^{4} - 25652 T^{6} + 1367 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 2 p T^{2} + 1775 T^{4} - 40932 T^{6} + 1775 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 29 | \( 1 - 66 T^{2} + 3207 T^{4} - 111228 T^{6} + 3207 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} \) |
| 31 | \( ( 1 - 8 T + 89 T^{2} - 432 T^{3} + 89 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 + 8 T + 3 p T^{2} + 584 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( ( 1 + 2 T + 23 T^{2} + 220 T^{3} + 23 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( ( 1 + 65 T^{2} - 64 T^{3} + 65 p T^{4} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 222 T^{2} + 22367 T^{4} - 1328324 T^{6} + 22367 p^{2} T^{8} - 222 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 - 12 T + 191 T^{2} - 1264 T^{3} + 191 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 178 T^{2} + 20567 T^{4} - 1418652 T^{6} + 20567 p^{2} T^{8} - 178 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 - 190 T^{2} + 20039 T^{4} - 1419204 T^{6} + 20039 p^{2} T^{8} - 190 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 + 137 T^{2} + 64 T^{3} + 137 p T^{4} + p^{3} T^{6} )^{2} \) |
| 71 | \( ( 1 + 8 T + 133 T^{2} + 1008 T^{3} + 133 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 54 T^{2} + 2367 T^{4} - 531700 T^{6} + 2367 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 8 T + 233 T^{2} + 1200 T^{3} + 233 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( ( 1 - 8 T + 185 T^{2} - 880 T^{3} + 185 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( ( 1 + 10 T + 103 T^{2} + 396 T^{3} + 103 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 246 T^{2} + 39183 T^{4} - 4535476 T^{6} + 39183 p^{2} T^{8} - 246 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−5.97662037124656368051745290114, −5.75199166769092078394248540984, −5.63670131398747127111629501554, −5.34199320992128241234921129174, −5.20655283360584712611943882292, −5.02681864634932828086060195998, −4.95931229340575094338685669197, −4.86058858165686218688797270128, −4.69339049838246513439474849954, −4.27143611009554372742365879392, −4.25101324916691323545673391185, −4.19464789542974790850354743528, −3.81865239981444901718684137271, −3.61845864906280522997966773554, −3.45478971363305508164919836662, −2.97841773126612876502055257952, −2.66392667684989614103597525436, −2.60908967188182126178160123972, −2.44669549968000715208754961176, −1.91634765867288201033262170352, −1.66927737273400304016758006046, −1.53387273057320446238445883456, −0.798591210914628693434766865238, −0.78857349215925599222326520489, −0.33163425643839558471229567552,
0.33163425643839558471229567552, 0.78857349215925599222326520489, 0.798591210914628693434766865238, 1.53387273057320446238445883456, 1.66927737273400304016758006046, 1.91634765867288201033262170352, 2.44669549968000715208754961176, 2.60908967188182126178160123972, 2.66392667684989614103597525436, 2.97841773126612876502055257952, 3.45478971363305508164919836662, 3.61845864906280522997966773554, 3.81865239981444901718684137271, 4.19464789542974790850354743528, 4.25101324916691323545673391185, 4.27143611009554372742365879392, 4.69339049838246513439474849954, 4.86058858165686218688797270128, 4.95931229340575094338685669197, 5.02681864634932828086060195998, 5.20655283360584712611943882292, 5.34199320992128241234921129174, 5.63670131398747127111629501554, 5.75199166769092078394248540984, 5.97662037124656368051745290114
Plot not available for L-functions of degree greater than 10.
