| L(s) = 1 | + 4-s − 8·13-s − 2·16-s − 8·17-s − 16·23-s − 3·25-s + 4·29-s + 24·43-s + 22·49-s − 8·52-s − 12·53-s − 12·61-s + 6·64-s − 8·68-s + 24·79-s − 16·92-s − 3·100-s + 68·101-s − 28·103-s − 8·107-s + 8·113-s + 4·116-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 2.21·13-s − 1/2·16-s − 1.94·17-s − 3.33·23-s − 3/5·25-s + 0.742·29-s + 3.65·43-s + 22/7·49-s − 1.10·52-s − 1.64·53-s − 1.53·61-s + 3/4·64-s − 0.970·68-s + 2.70·79-s − 1.66·92-s − 0.299·100-s + 6.76·101-s − 2.75·103-s − 0.773·107-s + 0.752·113-s + 0.371·116-s + 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.494090841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.494090841\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 + T^{2} )^{3} \) |
| 13 | \( 1 + 8 T + 3 p T^{2} + 160 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 2 | \( 1 - T^{2} + 3 T^{4} - 11 T^{6} + 3 p^{2} T^{8} - p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 22 T^{2} + 41 p T^{4} - 2404 T^{6} + 41 p^{3} T^{8} - 22 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 18 T^{2} + 387 T^{4} - 3904 T^{6} + 387 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 + 4 T + 19 T^{2} + 40 T^{3} + 19 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( 1 - 70 T^{2} + 2267 T^{4} - 49288 T^{6} + 2267 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 + 8 T + 85 T^{2} + 374 T^{3} + 85 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 - 2 T + 79 T^{2} - 104 T^{3} + 79 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 130 T^{2} + 8075 T^{4} - 310336 T^{6} + 8075 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( 1 - 70 T^{2} + 4295 T^{4} - 175492 T^{6} + 4295 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 154 T^{2} + 12351 T^{4} - 627692 T^{6} + 12351 p^{2} T^{8} - 154 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 - 12 T + 141 T^{2} - 1034 T^{3} + 141 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 262 T^{2} + 29487 T^{4} - 1821764 T^{6} + 29487 p^{2} T^{8} - 262 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 + 6 T + 99 T^{2} + 708 T^{3} + 99 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 270 T^{2} + 32859 T^{4} - 2408056 T^{6} + 32859 p^{2} T^{8} - 270 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 + 6 T + 171 T^{2} + 656 T^{3} + 171 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 246 T^{2} + 28695 T^{4} - 2226404 T^{6} + 28695 p^{2} T^{8} - 246 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( 1 - 382 T^{2} + 63315 T^{4} - 5855192 T^{6} + 63315 p^{2} T^{8} - 382 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( 1 - 142 T^{2} + 16847 T^{4} - 1458004 T^{6} + 16847 p^{2} T^{8} - 142 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 12 T + 189 T^{2} - 1864 T^{3} + 189 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 342 T^{2} + 57303 T^{4} - 5942500 T^{6} + 57303 p^{2} T^{8} - 342 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 - 342 T^{2} + 59679 T^{4} - 6531892 T^{6} + 59679 p^{2} T^{8} - 342 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 - 442 T^{2} + 88271 T^{4} - 10631788 T^{6} + 88271 p^{2} T^{8} - 442 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−6.01697583541036363858513577365, −5.44721569294066927441226191490, −5.42589544869629850860401160514, −5.25315172997046177771128233011, −4.95573894289646753833313455867, −4.84598598611753600422236486246, −4.80235020557715445685897616637, −4.52352745621597993354894374339, −4.16957085826579054765720245161, −4.03042817275080708270884744591, −4.00740433812267806814923646339, −3.97719157037742959725082459355, −3.84379175514867504841765328702, −3.20821774991765613122317316101, −3.10016418996427188054326126835, −2.76202744438686352046787282656, −2.70949790635949699963224549918, −2.42977665878316321708026258175, −2.13368966717592242971619745929, −2.12831735536888643355156915966, −1.92438513430961174290774650621, −1.81133743791115713385531021838, −1.07320586777369115378600175713, −0.63090116261909662255959524027, −0.32832438382575718081671312625,
0.32832438382575718081671312625, 0.63090116261909662255959524027, 1.07320586777369115378600175713, 1.81133743791115713385531021838, 1.92438513430961174290774650621, 2.12831735536888643355156915966, 2.13368966717592242971619745929, 2.42977665878316321708026258175, 2.70949790635949699963224549918, 2.76202744438686352046787282656, 3.10016418996427188054326126835, 3.20821774991765613122317316101, 3.84379175514867504841765328702, 3.97719157037742959725082459355, 4.00740433812267806814923646339, 4.03042817275080708270884744591, 4.16957085826579054765720245161, 4.52352745621597993354894374339, 4.80235020557715445685897616637, 4.84598598611753600422236486246, 4.95573894289646753833313455867, 5.25315172997046177771128233011, 5.42589544869629850860401160514, 5.44721569294066927441226191490, 6.01697583541036363858513577365
Plot not available for L-functions of degree greater than 10.
