| L(s) = 1 | + 15·5-s − 15·7-s + 24·11-s − 33·13-s − 84·17-s + 42·19-s − 33·23-s + 75·25-s + 222·29-s − 132·31-s − 225·35-s + 348·37-s − 99·41-s + 120·43-s + 537·47-s + 381·49-s − 534·53-s + 360·55-s − 225·59-s + 480·61-s − 495·65-s − 12·67-s − 1.14e3·71-s + 2.12e3·73-s − 360·77-s + 1.02e3·79-s + 702·83-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.809·7-s + 0.657·11-s − 0.704·13-s − 1.19·17-s + 0.507·19-s − 0.299·23-s + 3/5·25-s + 1.42·29-s − 0.764·31-s − 1.08·35-s + 1.54·37-s − 0.377·41-s + 0.425·43-s + 1.66·47-s + 1.11·49-s − 1.38·53-s + 0.882·55-s − 0.496·59-s + 1.00·61-s − 0.944·65-s − 0.0218·67-s − 1.90·71-s + 3.40·73-s − 0.532·77-s + 1.46·79-s + 0.928·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(19.87526019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(19.87526019\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 - p T + p^{2} T^{2} )^{3} \) |
| good | 7 | \( 1 + 15 T - 156 T^{2} - 13609 T^{3} - 18636 p T^{4} + 1525467 T^{5} + 97800210 T^{6} + 1525467 p^{3} T^{7} - 18636 p^{7} T^{8} - 13609 p^{9} T^{9} - 156 p^{12} T^{10} + 15 p^{15} T^{11} + p^{18} T^{12} \) |
| 11 | \( 1 - 24 T - 1272 T^{2} + 118128 T^{3} - 994344 T^{4} - 6780360 p T^{5} + 5749910422 T^{6} - 6780360 p^{4} T^{7} - 994344 p^{6} T^{8} + 118128 p^{9} T^{9} - 1272 p^{12} T^{10} - 24 p^{15} T^{11} + p^{18} T^{12} \) |
| 13 | \( 1 + 33 T - 5142 T^{2} - 76861 T^{3} + 20474868 T^{4} + 135856137 T^{5} - 49397411808 T^{6} + 135856137 p^{3} T^{7} + 20474868 p^{6} T^{8} - 76861 p^{9} T^{9} - 5142 p^{12} T^{10} + 33 p^{15} T^{11} + p^{18} T^{12} \) |
| 17 | \( ( 1 + 42 T + 3855 T^{2} - 16788 T^{3} + 3855 p^{3} T^{4} + 42 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 19 | \( ( 1 - 21 T + 12456 T^{2} + 23647 T^{3} + 12456 p^{3} T^{4} - 21 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 23 | \( 1 + 33 T - 816 p T^{2} - 2049699 T^{3} + 108068160 T^{4} + 15795970413 T^{5} + 472869876922 T^{6} + 15795970413 p^{3} T^{7} + 108068160 p^{6} T^{8} - 2049699 p^{9} T^{9} - 816 p^{13} T^{10} + 33 p^{15} T^{11} + p^{18} T^{12} \) |
| 29 | \( 1 - 222 T - 14808 T^{2} + 6723696 T^{3} + 218959356 T^{4} - 109448348910 T^{5} + 9262097230678 T^{6} - 109448348910 p^{3} T^{7} + 218959356 p^{6} T^{8} + 6723696 p^{9} T^{9} - 14808 p^{12} T^{10} - 222 p^{15} T^{11} + p^{18} T^{12} \) |
| 31 | \( 1 + 132 T - 52680 T^{2} - 4195360 T^{3} + 1938462744 T^{4} + 37212296772 T^{5} - 63775106987922 T^{6} + 37212296772 p^{3} T^{7} + 1938462744 p^{6} T^{8} - 4195360 p^{9} T^{9} - 52680 p^{12} T^{10} + 132 p^{15} T^{11} + p^{18} T^{12} \) |
| 37 | \( ( 1 - 174 T + 103359 T^{2} - 18477164 T^{3} + 103359 p^{3} T^{4} - 174 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 41 | \( 1 + 99 T - 174201 T^{2} - 6194160 T^{3} + 19966821441 T^{4} + 205080501501 T^{5} - 1576265610587762 T^{6} + 205080501501 p^{3} T^{7} + 19966821441 p^{6} T^{8} - 6194160 p^{9} T^{9} - 174201 p^{12} T^{10} + 99 p^{15} T^{11} + p^{18} T^{12} \) |
| 43 | \( 1 - 120 T - 144201 T^{2} - 6468760 T^{3} + 11790585894 T^{4} + 1271529234120 T^{5} - 1046324017829421 T^{6} + 1271529234120 p^{3} T^{7} + 11790585894 p^{6} T^{8} - 6468760 p^{9} T^{9} - 144201 p^{12} T^{10} - 120 p^{15} T^{11} + p^{18} T^{12} \) |
| 47 | \( 1 - 537 T - 104520 T^{2} + 18225291 T^{3} + 52028857020 T^{4} - 7005388735833 T^{5} - 3762506266179086 T^{6} - 7005388735833 p^{3} T^{7} + 52028857020 p^{6} T^{8} + 18225291 p^{9} T^{9} - 104520 p^{12} T^{10} - 537 p^{15} T^{11} + p^{18} T^{12} \) |
| 53 | \( ( 1 + 267 T + 3687 p T^{2} + 93575922 T^{3} + 3687 p^{4} T^{4} + 267 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 59 | \( 1 + 225 T - 366375 T^{2} - 65529378 T^{3} + 70318589475 T^{4} + 4176638532225 T^{5} - 14174092362324026 T^{6} + 4176638532225 p^{3} T^{7} + 70318589475 p^{6} T^{8} - 65529378 p^{9} T^{9} - 366375 p^{12} T^{10} + 225 p^{15} T^{11} + p^{18} T^{12} \) |
| 61 | \( 1 - 480 T - 150963 T^{2} + 168903680 T^{3} - 25594318134 T^{4} - 8741974916880 T^{5} + 9474989139030093 T^{6} - 8741974916880 p^{3} T^{7} - 25594318134 p^{6} T^{8} + 168903680 p^{9} T^{9} - 150963 p^{12} T^{10} - 480 p^{15} T^{11} + p^{18} T^{12} \) |
| 67 | \( 1 + 12 T - 417693 T^{2} + 108253292 T^{3} + 49537284426 T^{4} - 23499166078404 T^{5} + 4555090336970787 T^{6} - 23499166078404 p^{3} T^{7} + 49537284426 p^{6} T^{8} + 108253292 p^{9} T^{9} - 417693 p^{12} T^{10} + 12 p^{15} T^{11} + p^{18} T^{12} \) |
| 71 | \( ( 1 + 570 T + 1134960 T^{2} + 5765802 p T^{3} + 1134960 p^{3} T^{4} + 570 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( ( 1 - 1062 T + 1310247 T^{2} - 746703380 T^{3} + 1310247 p^{3} T^{4} - 1062 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 79 | \( 1 - 1026 T - 271161 T^{2} + 139008254 T^{3} + 331744178718 T^{4} + 98871968501694 T^{5} - 319811246026634685 T^{6} + 98871968501694 p^{3} T^{7} + 331744178718 p^{6} T^{8} + 139008254 p^{9} T^{9} - 271161 p^{12} T^{10} - 1026 p^{15} T^{11} + p^{18} T^{12} \) |
| 83 | \( 1 - 702 T + 123 T^{2} + 308296314 T^{3} - 268583613162 T^{4} + 4861018161522 T^{5} + 180951311703724567 T^{6} + 4861018161522 p^{3} T^{7} - 268583613162 p^{6} T^{8} + 308296314 p^{9} T^{9} + 123 p^{12} T^{10} - 702 p^{15} T^{11} + p^{18} T^{12} \) |
| 89 | \( ( 1 + 1140 T + 2294160 T^{2} + 1604869674 T^{3} + 2294160 p^{3} T^{4} + 1140 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 97 | \( 1 - 2556 T + 2105517 T^{2} - 1390881412 T^{3} + 2877269596518 T^{4} - 3446169123766524 T^{5} + 2818603223778559113 T^{6} - 3446169123766524 p^{3} T^{7} + 2877269596518 p^{6} T^{8} - 1390881412 p^{9} T^{9} + 2105517 p^{12} T^{10} - 2556 p^{15} T^{11} + p^{18} T^{12} \) |
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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
−4.61198794326396129404767301557, −4.26692996345239900189696626756, −4.15418770964063357486388464421, −4.14438077351539815307842384329, −4.07414996132301543674937289127, −3.77837751325313850283036367636, −3.38782212027835013437355620159, −3.30327528166789611901026515796, −3.18058234027122998657035786221, −3.14187768928383184646489084073, −2.97875498651199060523819692112, −2.76303431991399633200385297187, −2.45422041817290260652168076528, −2.35284044220657609937701660252, −2.05261690036751779024700795583, −1.97487690371201575242372545517, −1.85637039294700274950010712844, −1.82652389452497372935960862269, −1.65661640727364355812307616483, −1.01348053581673737073261856465, −0.894125300222733377239505084064, −0.70200045830276321285471631959, −0.65213445867591631333707162639, −0.57112005679271706943456757128, −0.24518963504275157800621430020,
0.24518963504275157800621430020, 0.57112005679271706943456757128, 0.65213445867591631333707162639, 0.70200045830276321285471631959, 0.894125300222733377239505084064, 1.01348053581673737073261856465, 1.65661640727364355812307616483, 1.82652389452497372935960862269, 1.85637039294700274950010712844, 1.97487690371201575242372545517, 2.05261690036751779024700795583, 2.35284044220657609937701660252, 2.45422041817290260652168076528, 2.76303431991399633200385297187, 2.97875498651199060523819692112, 3.14187768928383184646489084073, 3.18058234027122998657035786221, 3.30327528166789611901026515796, 3.38782212027835013437355620159, 3.77837751325313850283036367636, 4.07414996132301543674937289127, 4.14438077351539815307842384329, 4.15418770964063357486388464421, 4.26692996345239900189696626756, 4.61198794326396129404767301557
Plot not available for L-functions of degree greater than 10.
