L(s) = 1 | + 3·2-s + 3·3-s + 6·4-s + 3·5-s + 9·6-s + 9·8-s + 9·9-s + 9·10-s + 18·12-s + 9·15-s + 12·16-s − 6·17-s + 27·18-s + 18·20-s + 6·23-s + 27·24-s + 15·25-s + 20·27-s + 15·29-s + 27·30-s − 18·31-s + 12·32-s − 18·34-s + 54·36-s + 27·40-s + 12·41-s + 27·45-s + ⋯ |
L(s) = 1 | + 2.12·2-s + 1.73·3-s + 3·4-s + 1.34·5-s + 3.67·6-s + 3.18·8-s + 3·9-s + 2.84·10-s + 5.19·12-s + 2.32·15-s + 3·16-s − 1.45·17-s + 6.36·18-s + 4.02·20-s + 1.25·23-s + 5.51·24-s + 3·25-s + 3.84·27-s + 2.78·29-s + 4.92·30-s − 3.23·31-s + 2.12·32-s − 3.08·34-s + 9·36-s + 4.26·40-s + 1.87·41-s + 4.02·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(39.38713552\) |
\(L(\frac12)\) |
\(\approx\) |
\(39.38713552\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 19 | \( 1 \) |
good | 2 | \( 1 - 3 T + 3 T^{2} - 3 T^{4} + 3 p T^{5} - 11 T^{6} + 3 p^{2} T^{7} - 3 p^{2} T^{8} + 3 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 3 | \( 1 - p T + 7 T^{3} + p T^{4} - 2 p^{2} T^{5} + 19 T^{6} - 2 p^{3} T^{7} + p^{3} T^{8} + 7 p^{3} T^{9} - p^{6} T^{11} + p^{6} T^{12} \) |
| 5 | \( 1 - 3 T - 6 T^{2} + 9 T^{3} + 69 T^{4} - 6 p T^{5} - 371 T^{6} - 6 p^{2} T^{7} + 69 p^{2} T^{8} + 9 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 7 | \( ( 1 + 18 T^{2} + T^{3} + 18 p T^{4} + p^{3} T^{6} )^{2} \) |
| 11 | \( ( 1 + 24 T^{2} - 9 T^{3} + 24 p T^{4} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 18 T^{2} + 74 T^{3} + 90 T^{4} - 666 T^{5} + 1551 T^{6} - 666 p T^{7} + 90 p^{2} T^{8} + 74 p^{3} T^{9} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 + 6 T - 24 T^{2} - 54 T^{3} + 1338 T^{4} + 1914 T^{5} - 18929 T^{6} + 1914 p T^{7} + 1338 p^{2} T^{8} - 54 p^{3} T^{9} - 24 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 - 6 T - 33 T^{2} + 90 T^{3} + 1662 T^{4} - 66 p T^{5} - 40709 T^{6} - 66 p^{2} T^{7} + 1662 p^{2} T^{8} + 90 p^{3} T^{9} - 33 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 - 15 T + 66 T^{2} - 423 T^{3} + 5955 T^{4} - 30264 T^{5} + 88765 T^{6} - 30264 p T^{7} + 5955 p^{2} T^{8} - 423 p^{3} T^{9} + 66 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( ( 1 + 9 T + 99 T^{2} + 505 T^{3} + 99 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 + 90 T^{2} - 17 T^{3} + 90 p T^{4} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 - 12 T + 12 T^{2} + 162 T^{3} + 1536 T^{4} - 5520 T^{5} - 56477 T^{6} - 5520 p T^{7} + 1536 p^{2} T^{8} + 162 p^{3} T^{9} + 12 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 - 72 T^{2} + 326 T^{3} + 2088 T^{4} - 11736 T^{5} - 37329 T^{6} - 11736 p T^{7} + 2088 p^{2} T^{8} + 326 p^{3} T^{9} - 72 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 6 T - 96 T^{2} + 342 T^{3} + 7818 T^{4} - 13650 T^{5} - 369317 T^{6} - 13650 p T^{7} + 7818 p^{2} T^{8} + 342 p^{3} T^{9} - 96 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 - 6 T - 114 T^{2} + 270 T^{3} + 11040 T^{4} - 9204 T^{5} - 653645 T^{6} - 9204 p T^{7} + 11040 p^{2} T^{8} + 270 p^{3} T^{9} - 114 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 - 21 T + 129 T^{2} - 18 p T^{3} + 21291 T^{4} - 156405 T^{5} + 667366 T^{6} - 156405 p T^{7} + 21291 p^{2} T^{8} - 18 p^{4} T^{9} + 129 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 + 9 T - 81 T^{2} - 376 T^{3} + 8109 T^{4} + 4167 T^{5} - 607434 T^{6} + 4167 p T^{7} + 8109 p^{2} T^{8} - 376 p^{3} T^{9} - 81 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 + 18 T + 99 T^{2} + 74 T^{3} - 234 T^{4} - 14598 T^{5} - 292641 T^{6} - 14598 p T^{7} - 234 p^{2} T^{8} + 74 p^{3} T^{9} + 99 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 30 T + 399 T^{2} - 4734 T^{3} + 60990 T^{4} - 596118 T^{5} + 4859755 T^{6} - 596118 p T^{7} + 60990 p^{2} T^{8} - 4734 p^{3} T^{9} + 399 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 - 171 T^{2} + 128 T^{3} + 16758 T^{4} - 10944 T^{5} - 1352463 T^{6} - 10944 p T^{7} + 16758 p^{2} T^{8} + 128 p^{3} T^{9} - 171 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( 1 - 9 T - 54 T^{2} + 11 T^{3} + 2043 T^{4} + 53046 T^{5} - 583449 T^{6} + 53046 p T^{7} + 2043 p^{2} T^{8} + 11 p^{3} T^{9} - 54 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( ( 1 + 60 T^{2} + 459 T^{3} + 60 p T^{4} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 - 15 T - 96 T^{2} + 639 T^{3} + 33567 T^{4} - 137112 T^{5} - 2106407 T^{6} - 137112 p T^{7} + 33567 p^{2} T^{8} + 639 p^{3} T^{9} - 96 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 + 15 T - 105 T^{2} - 616 T^{3} + 35145 T^{4} + 99225 T^{5} - 3109890 T^{6} + 99225 p T^{7} + 35145 p^{2} T^{8} - 616 p^{3} T^{9} - 105 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−6.24953451420290035858520066245, −6.18166093914630067042616348577, −5.77375899010801672511440655986, −5.60380584396947447872565521286, −5.44521634149830749700769607079, −5.06899967755467811027229411170, −5.00163610972480703962402769544, −4.89814154157425354133355924414, −4.76688995845233362455981196489, −4.71548747809601184385838643750, −4.29286014471803721553004857802, −4.06820362547782848653494426733, −3.74804974754259904503006531201, −3.66739364071625099927607414518, −3.66717510290389246287173959939, −3.38519265948814202718072197775, −2.84715859889982907217347872798, −2.72187469338323217796684481967, −2.64752170959967492202754172550, −2.48358848480445307916634717698, −2.36536658173243725557177759024, −1.80354922196562680540283447645, −1.52187997528181881684529123810, −1.40266350574039103710252165398, −1.05239814183596614313126055052,
1.05239814183596614313126055052, 1.40266350574039103710252165398, 1.52187997528181881684529123810, 1.80354922196562680540283447645, 2.36536658173243725557177759024, 2.48358848480445307916634717698, 2.64752170959967492202754172550, 2.72187469338323217796684481967, 2.84715859889982907217347872798, 3.38519265948814202718072197775, 3.66717510290389246287173959939, 3.66739364071625099927607414518, 3.74804974754259904503006531201, 4.06820362547782848653494426733, 4.29286014471803721553004857802, 4.71548747809601184385838643750, 4.76688995845233362455981196489, 4.89814154157425354133355924414, 5.00163610972480703962402769544, 5.06899967755467811027229411170, 5.44521634149830749700769607079, 5.60380584396947447872565521286, 5.77375899010801672511440655986, 6.18166093914630067042616348577, 6.24953451420290035858520066245
Plot not available for L-functions of degree greater than 10.