| L(s) = 1 | + 24·7-s + 12·13-s + 258·19-s + 2.14e3·25-s + 2.58e3·31-s + 12·37-s − 570·43-s − 5.05e3·49-s − 7.26e3·61-s − 1.01e4·67-s − 1.46e4·73-s + 9.52e3·79-s + 288·91-s + 5.79e4·97-s − 1.31e4·103-s − 1.15e4·109-s + 5.96e4·121-s + 127-s + 131-s + 6.19e3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.489·7-s + 0.0710·13-s + 0.714·19-s + 3.43·25-s + 2.68·31-s + 0.00876·37-s − 0.308·43-s − 2.10·49-s − 1.95·61-s − 2.25·67-s − 2.74·73-s + 1.52·79-s + 0.0347·91-s + 6.15·97-s − 1.24·103-s − 0.971·109-s + 4.07·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 0.350·133-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(8.042414166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.042414166\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2148 T^{2} + 473448 p T^{4} - 1726204862 T^{6} + 473448 p^{9} T^{8} - 2148 p^{16} T^{10} + p^{24} T^{12} \) |
| 7 | \( ( 1 - 12 T + 2742 T^{2} + 62894 T^{3} + 2742 p^{4} T^{4} - 12 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 11 | \( 1 - 59649 T^{2} + 1791330522 T^{4} - 32676716841545 T^{6} + 1791330522 p^{8} T^{8} - 59649 p^{16} T^{10} + p^{24} T^{12} \) |
| 13 | \( ( 1 - 6 T + 68388 T^{2} - 1184384 T^{3} + 68388 p^{4} T^{4} - 6 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 17 | \( 1 - 346011 T^{2} + 53617620939 T^{4} - 5294214329064626 T^{6} + 53617620939 p^{8} T^{8} - 346011 p^{16} T^{10} + p^{24} T^{12} \) |
| 19 | \( ( 1 - 129 T + 372939 T^{2} - 32427790 T^{3} + 372939 p^{4} T^{4} - 129 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 23 | \( 1 - 1218228 T^{2} + 707030235312 T^{4} - 248730025998307322 T^{6} + 707030235312 p^{8} T^{8} - 1218228 p^{16} T^{10} + p^{24} T^{12} \) |
| 29 | \( 1 - 2821668 T^{2} + 4127228848680 T^{4} - 3630313408885280318 T^{6} + 4127228848680 p^{8} T^{8} - 2821668 p^{16} T^{10} + p^{24} T^{12} \) |
| 31 | \( ( 1 - 1290 T + 2234706 T^{2} - 2130154468 T^{3} + 2234706 p^{4} T^{4} - 1290 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 37 | \( ( 1 - 6 T + 3399531 T^{2} + 1254253444 T^{3} + 3399531 p^{4} T^{4} - 6 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 41 | \( 1 - 8356461 T^{2} + 35786746147974 T^{4} - \)\(11\!\cdots\!65\)\( T^{6} + 35786746147974 p^{8} T^{8} - 8356461 p^{16} T^{10} + p^{24} T^{12} \) |
| 43 | \( ( 1 + 285 T + 4498080 T^{2} - 2707378441 T^{3} + 4498080 p^{4} T^{4} + 285 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 47 | \( 1 - 12334308 T^{2} + 47182743741552 T^{4} - 98872971142023729002 T^{6} + 47182743741552 p^{8} T^{8} - 12334308 p^{16} T^{10} + p^{24} T^{12} \) |
| 53 | \( 1 - 20649822 T^{2} + 267970470920223 T^{4} - \)\(22\!\cdots\!72\)\( T^{6} + 267970470920223 p^{8} T^{8} - 20649822 p^{16} T^{10} + p^{24} T^{12} \) |
| 59 | \( 1 - 29985513 T^{2} + 550138778452554 T^{4} - \)\(73\!\cdots\!17\)\( T^{6} + 550138778452554 p^{8} T^{8} - 29985513 p^{16} T^{10} + p^{24} T^{12} \) |
| 61 | \( ( 1 + 3630 T + 44849796 T^{2} + 100569730372 T^{3} + 44849796 p^{4} T^{4} + 3630 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 67 | \( ( 1 + 5055 T + 38905080 T^{2} + 95477585141 T^{3} + 38905080 p^{4} T^{4} + 5055 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 71 | \( 1 - 87967842 T^{2} + 4070274316914543 T^{4} - \)\(12\!\cdots\!12\)\( T^{6} + 4070274316914543 p^{8} T^{8} - 87967842 p^{16} T^{10} + p^{24} T^{12} \) |
| 73 | \( ( 1 + 7311 T + 93929019 T^{2} + 399497247430 T^{3} + 93929019 p^{4} T^{4} + 7311 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 79 | \( ( 1 - 4764 T + 111036894 T^{2} - 369476578510 T^{3} + 111036894 p^{4} T^{4} - 4764 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 83 | \( 1 - 17707524 T^{2} + 20420433937104 T^{4} - \)\(16\!\cdots\!94\)\( T^{6} + 20420433937104 p^{8} T^{8} - 17707524 p^{16} T^{10} + p^{24} T^{12} \) |
| 89 | \( 1 - 92525118 T^{2} + 4141804686688959 T^{4} - \)\(27\!\cdots\!72\)\( T^{6} + 4141804686688959 p^{8} T^{8} - 92525118 p^{16} T^{10} + p^{24} T^{12} \) |
| 97 | \( ( 1 - 28959 T + 522623814 T^{2} - 5850256931075 T^{3} + 522623814 p^{4} T^{4} - 28959 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
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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
−4.62050558125584099414358526361, −4.53367274649792542146443536606, −4.48618923443973612501139920325, −3.94309479581978942882794924851, −3.75690782835403294379042801723, −3.62793108790167369604099179008, −3.51960641260163592194384492173, −3.20993056884104875664987040177, −3.19741831527829844597734480194, −3.03901251843940475758832853205, −2.93796611606663355729874219580, −2.63768889757575290163255496457, −2.60860625150699030442978122356, −2.31579799268157651734015535985, −2.13212818577923400884185404776, −1.86871092172126808560526891795, −1.76137388649450411268363777048, −1.44021390928757119958994332191, −1.31988882234329695579897577441, −1.08076036736390509993263442631, −1.05628879084602569688587383570, −0.833244027289874798280417489511, −0.51226952419249827361436490443, −0.41440921846529108194341980701, −0.16056736074127671409665439072,
0.16056736074127671409665439072, 0.41440921846529108194341980701, 0.51226952419249827361436490443, 0.833244027289874798280417489511, 1.05628879084602569688587383570, 1.08076036736390509993263442631, 1.31988882234329695579897577441, 1.44021390928757119958994332191, 1.76137388649450411268363777048, 1.86871092172126808560526891795, 2.13212818577923400884185404776, 2.31579799268157651734015535985, 2.60860625150699030442978122356, 2.63768889757575290163255496457, 2.93796611606663355729874219580, 3.03901251843940475758832853205, 3.19741831527829844597734480194, 3.20993056884104875664987040177, 3.51960641260163592194384492173, 3.62793108790167369604099179008, 3.75690782835403294379042801723, 3.94309479581978942882794924851, 4.48618923443973612501139920325, 4.53367274649792542146443536606, 4.62050558125584099414358526361
Plot not available for L-functions of degree greater than 10.
