| L(s) = 1 | − 2-s − 4·4-s + 5·5-s + 7-s + 3·8-s − 5·10-s + 2·11-s − 5·13-s − 14-s + 7·16-s + 10·17-s − 20·20-s − 2·22-s + 3·23-s + 5·26-s − 4·28-s + 6·29-s + 2·31-s − 32-s − 10·34-s + 5·35-s − 13·37-s + 15·40-s + 13·41-s + 3·43-s − 8·44-s − 3·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 2·4-s + 2.23·5-s + 0.377·7-s + 1.06·8-s − 1.58·10-s + 0.603·11-s − 1.38·13-s − 0.267·14-s + 7/4·16-s + 2.42·17-s − 4.47·20-s − 0.426·22-s + 0.625·23-s + 0.980·26-s − 0.755·28-s + 1.11·29-s + 0.359·31-s − 0.176·32-s − 1.71·34-s + 0.845·35-s − 2.13·37-s + 2.37·40-s + 2.03·41-s + 0.457·43-s − 1.20·44-s − 0.442·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.71467773\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.71467773\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + T + 5 T^{2} + 3 p T^{3} + p^{4} T^{4} + 19 T^{5} + 37 T^{6} + 19 p T^{7} + p^{6} T^{8} + 3 p^{4} T^{9} + 5 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \) |
| 5 | \( 1 - p T + p^{2} T^{2} - 89 T^{3} + 284 T^{4} - 758 T^{5} + 1849 T^{6} - 758 p T^{7} + 284 p^{2} T^{8} - 89 p^{3} T^{9} + p^{6} T^{10} - p^{6} T^{11} + p^{6} T^{12} \) |
| 7 | \( 1 - T + 15 T^{2} - T^{3} + 11 p T^{4} + 257 T^{5} + 297 T^{6} + 257 p T^{7} + 11 p^{3} T^{8} - p^{3} T^{9} + 15 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 - 2 T + 39 T^{2} - 104 T^{3} + 816 T^{4} - 1979 T^{5} + 11257 T^{6} - 1979 p T^{7} + 816 p^{2} T^{8} - 104 p^{3} T^{9} + 39 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( 1 + 5 T + 48 T^{2} + 198 T^{3} + 1120 T^{4} + 3623 T^{5} + 1327 p T^{6} + 3623 p T^{7} + 1120 p^{2} T^{8} + 198 p^{3} T^{9} + 48 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 - 10 T + 108 T^{2} - 730 T^{3} + 4647 T^{4} - 23203 T^{5} + 105427 T^{6} - 23203 p T^{7} + 4647 p^{2} T^{8} - 730 p^{3} T^{9} + 108 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 - 3 T + 102 T^{2} - 340 T^{3} + 4806 T^{4} - 15537 T^{5} + 137275 T^{6} - 15537 p T^{7} + 4806 p^{2} T^{8} - 340 p^{3} T^{9} + 102 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 - 6 T + 108 T^{2} - 643 T^{3} + 6189 T^{4} - 30039 T^{5} + 225151 T^{6} - 30039 p T^{7} + 6189 p^{2} T^{8} - 643 p^{3} T^{9} + 108 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( 1 - 2 T + 115 T^{2} - 335 T^{3} + 6495 T^{4} - 20466 T^{5} + 241509 T^{6} - 20466 p T^{7} + 6495 p^{2} T^{8} - 335 p^{3} T^{9} + 115 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( 1 + 13 T + 166 T^{2} + 1471 T^{3} + 12180 T^{4} + 80949 T^{5} + 541779 T^{6} + 80949 p T^{7} + 12180 p^{2} T^{8} + 1471 p^{3} T^{9} + 166 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \) |
| 41 | \( 1 - 13 T + 301 T^{2} - 2731 T^{3} + 34163 T^{4} - 226519 T^{5} + 1927273 T^{6} - 226519 p T^{7} + 34163 p^{2} T^{8} - 2731 p^{3} T^{9} + 301 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 - 3 T + 180 T^{2} - 664 T^{3} + 15009 T^{4} - 57828 T^{5} + 782877 T^{6} - 57828 p T^{7} + 15009 p^{2} T^{8} - 664 p^{3} T^{9} + 180 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 - 18 T + 321 T^{2} - 3318 T^{3} + 33753 T^{4} - 252018 T^{5} + 1947427 T^{6} - 252018 p T^{7} + 33753 p^{2} T^{8} - 3318 p^{3} T^{9} + 321 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 - T + 168 T^{2} + 170 T^{3} + 15630 T^{4} + 19061 T^{5} + 1013653 T^{6} + 19061 p T^{7} + 15630 p^{2} T^{8} + 170 p^{3} T^{9} + 168 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 + 11 T + 201 T^{2} + 2501 T^{3} + 25239 T^{4} + 228155 T^{5} + 2006239 T^{6} + 228155 p T^{7} + 25239 p^{2} T^{8} + 2501 p^{3} T^{9} + 201 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 13 T + 302 T^{2} - 2937 T^{3} + 40903 T^{4} - 313964 T^{5} + 3198343 T^{6} - 313964 p T^{7} + 40903 p^{2} T^{8} - 2937 p^{3} T^{9} + 302 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 - 9 T + 285 T^{2} - 1865 T^{3} + 36141 T^{4} - 186171 T^{5} + 2888823 T^{6} - 186171 p T^{7} + 36141 p^{2} T^{8} - 1865 p^{3} T^{9} + 285 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 6 T + 273 T^{2} - 1326 T^{3} + 38085 T^{4} - 158838 T^{5} + 3349555 T^{6} - 158838 p T^{7} + 38085 p^{2} T^{8} - 1326 p^{3} T^{9} + 273 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 - 15 T + 336 T^{2} - 4101 T^{3} + 54297 T^{4} - 509478 T^{5} + 5129947 T^{6} - 509478 p T^{7} + 54297 p^{2} T^{8} - 4101 p^{3} T^{9} + 336 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 + 13 T + 345 T^{2} + 3697 T^{3} + 59423 T^{4} + 502693 T^{5} + 5945925 T^{6} + 502693 p T^{7} + 59423 p^{2} T^{8} + 3697 p^{3} T^{9} + 345 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 - 8 T + 127 T^{2} - 641 T^{3} + 7649 T^{4} + 8518 T^{5} - 17789 T^{6} + 8518 p T^{7} + 7649 p^{2} T^{8} - 641 p^{3} T^{9} + 127 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 + 29 T + 736 T^{2} + 12911 T^{3} + 190427 T^{4} + 2298644 T^{5} + 23646313 T^{6} + 2298644 p T^{7} + 190427 p^{2} T^{8} + 12911 p^{3} T^{9} + 736 p^{4} T^{10} + 29 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 - 20 T + 617 T^{2} - 7782 T^{3} + 136480 T^{4} - 1259071 T^{5} + 16609801 T^{6} - 1259071 p T^{7} + 136480 p^{2} T^{8} - 7782 p^{3} T^{9} + 617 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−3.97850262991389881937530015859, −3.61781567516580044201030161482, −3.53765091097467298176277643549, −3.52632381016622582928776515140, −3.46363706472867463649004142159, −3.23291601962328518998504382033, −2.99179472791759598116196344897, −2.93407235735261719980097866578, −2.81411472051132606849297289710, −2.79874272276272439687064349757, −2.44653529056035140032961203979, −2.18874587924506581964401047978, −2.18357140991363504981485766190, −2.16654137930229823688513105428, −1.98197913707749288271097853744, −1.72036054104962100415894731557, −1.58536512573221113519248173481, −1.55995631424377175925515859433, −1.29924968176210328509546775321, −1.03132032376112646507212180370, −0.930755535648229456026034830937, −0.63439903319207743788646431316, −0.51666406756193403789559319409, −0.46413359138784045889526148906, −0.36861468132550081678006057678,
0.36861468132550081678006057678, 0.46413359138784045889526148906, 0.51666406756193403789559319409, 0.63439903319207743788646431316, 0.930755535648229456026034830937, 1.03132032376112646507212180370, 1.29924968176210328509546775321, 1.55995631424377175925515859433, 1.58536512573221113519248173481, 1.72036054104962100415894731557, 1.98197913707749288271097853744, 2.16654137930229823688513105428, 2.18357140991363504981485766190, 2.18874587924506581964401047978, 2.44653529056035140032961203979, 2.79874272276272439687064349757, 2.81411472051132606849297289710, 2.93407235735261719980097866578, 2.99179472791759598116196344897, 3.23291601962328518998504382033, 3.46363706472867463649004142159, 3.52632381016622582928776515140, 3.53765091097467298176277643549, 3.61781567516580044201030161482, 3.97850262991389881937530015859
Plot not available for L-functions of degree greater than 10.
