| L(s) = 1 | + 33·4-s − 24·7-s + 12·13-s + 417·16-s − 258·19-s + 2.14e3·25-s − 792·28-s − 2.58e3·31-s + 12·37-s + 570·43-s − 5.05e3·49-s + 396·52-s − 7.26e3·61-s + 4.27e3·64-s + 1.01e4·67-s − 1.46e4·73-s − 8.51e3·76-s − 9.52e3·79-s − 288·91-s + 5.79e4·97-s + 7.08e4·100-s + 1.31e4·103-s − 1.15e4·109-s − 1.00e4·112-s + 5.96e4·121-s − 8.51e4·124-s + 127-s + ⋯ |
| L(s) = 1 | + 2.06·4-s − 0.489·7-s + 0.0710·13-s + 1.62·16-s − 0.714·19-s + 3.43·25-s − 1.01·28-s − 2.68·31-s + 0.00876·37-s + 0.308·43-s − 2.10·49-s + 0.146·52-s − 1.95·61-s + 1.04·64-s + 2.25·67-s − 2.74·73-s − 1.47·76-s − 1.52·79-s − 0.0347·91-s + 6.15·97-s + 7.08·100-s + 1.24·103-s − 0.971·109-s − 0.797·112-s + 4.07·121-s − 5.53·124-s + 6.20e−5·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24}\right)^{s/2} \, \Gamma_{\C}(s+2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.129873360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.129873360\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 33 T^{2} + 21 p^{5} T^{4} - 3173 p^{2} T^{6} + 21 p^{13} T^{8} - 33 p^{16} T^{10} + p^{24} T^{12} \) |
| 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
−7.23408934847508806152140726235, −6.98037494688114132909101626386, −6.90745367122069539216804855995, −6.85518625228457858012774093009, −6.41182843067272881239219189094, −6.19437257383548092607266841211, −6.15187988257844698873668249875, −5.94799913548585162747859723215, −5.51971611941578251668075768752, −5.30246450138377368325179605550, −4.90903019012486437490689258692, −4.80896385816320441334366807889, −4.52518872983607294374300415728, −4.28962100489138128727435418502, −3.54543008685051500150966938881, −3.52728455625387193470344408134, −3.37921181921254594760931101012, −2.85692070950817403172740062021, −2.78086085694427877982945905453, −2.24761194384899778885424542625, −2.06331651473712206522524538534, −1.72580961234895447035865275240, −1.26106192685789711683003144579, −0.897709996041028160588406124645, −0.18889668417741760637859008748,
0.18889668417741760637859008748, 0.897709996041028160588406124645, 1.26106192685789711683003144579, 1.72580961234895447035865275240, 2.06331651473712206522524538534, 2.24761194384899778885424542625, 2.78086085694427877982945905453, 2.85692070950817403172740062021, 3.37921181921254594760931101012, 3.52728455625387193470344408134, 3.54543008685051500150966938881, 4.28962100489138128727435418502, 4.52518872983607294374300415728, 4.80896385816320441334366807889, 4.90903019012486437490689258692, 5.30246450138377368325179605550, 5.51971611941578251668075768752, 5.94799913548585162747859723215, 6.15187988257844698873668249875, 6.19437257383548092607266841211, 6.41182843067272881239219189094, 6.85518625228457858012774093009, 6.90745367122069539216804855995, 6.98037494688114132909101626386, 7.23408934847508806152140726235
Plot not available for L-functions of degree greater than 10.
