| L(s) = 1 | − 4·3-s + 6·9-s + 24·23-s − 4·25-s + 16·29-s + 16·31-s + 8·47-s − 8·53-s − 24·59-s + 48·61-s − 48·67-s − 96·69-s + 48·73-s + 16·75-s − 24·79-s − 15·81-s − 64·87-s + 48·89-s − 64·93-s + 24·101-s − 24·107-s − 8·109-s − 24·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 2·9-s + 5.00·23-s − 4/5·25-s + 2.97·29-s + 2.87·31-s + 1.16·47-s − 1.09·53-s − 3.12·59-s + 6.14·61-s − 5.86·67-s − 11.5·69-s + 5.61·73-s + 1.84·75-s − 2.70·79-s − 5/3·81-s − 6.86·87-s + 5.08·89-s − 6.63·93-s + 2.38·101-s − 2.32·107-s − 0.766·109-s − 2.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.690769461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.690769461\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T + T^{2} )^{4} \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 4 T^{2} + 12 T^{4} - 24 T^{5} + 8 T^{6} + 96 T^{7} - 49 T^{8} + 96 p T^{9} + 8 p^{2} T^{10} - 24 p^{3} T^{11} + 12 p^{4} T^{12} + 4 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 + 24 T^{2} + 294 T^{4} - 624 T^{5} + 1728 T^{6} - 13536 T^{7} + 3635 T^{8} - 13536 p T^{9} + 1728 p^{2} T^{10} - 624 p^{3} T^{11} + 294 p^{4} T^{12} + 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( ( 1 - 32 T^{2} + 592 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 + 36 T^{2} + 732 T^{4} + 3192 T^{5} + 8904 T^{6} + 94752 T^{7} + 19967 T^{8} + 94752 p T^{9} + 8904 p^{2} T^{10} + 3192 p^{3} T^{11} + 732 p^{4} T^{12} + 36 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( 1 - 36 T^{2} + 192 T^{3} + 706 T^{4} - 5280 T^{5} + 13968 T^{6} + 74688 T^{7} - 425181 T^{8} + 74688 p T^{9} + 13968 p^{2} T^{10} - 5280 p^{3} T^{11} + 706 p^{4} T^{12} + 192 p^{5} T^{13} - 36 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( 1 - 24 T + 336 T^{2} - 3456 T^{3} + 28614 T^{4} - 8712 p T^{5} + 1230336 T^{6} - 294456 p T^{7} + 33940163 T^{8} - 294456 p^{2} T^{9} + 1230336 p^{2} T^{10} - 8712 p^{4} T^{11} + 28614 p^{4} T^{12} - 3456 p^{5} T^{13} + 336 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 8 T + 112 T^{2} - 664 T^{3} + 4842 T^{4} - 664 p T^{5} + 112 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 - 16 T + 76 T^{2} - 224 T^{3} + 2738 T^{4} - 10960 T^{5} - 63664 T^{6} + 402352 T^{7} - 682637 T^{8} + 402352 p T^{9} - 63664 p^{2} T^{10} - 10960 p^{3} T^{11} + 2738 p^{4} T^{12} - 224 p^{5} T^{13} + 76 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) |
| 37 | \( 1 - 48 T^{2} + 384 T^{3} + 526 T^{4} - 16320 T^{5} + 82944 T^{6} + 340608 T^{7} - 3231021 T^{8} + 340608 p T^{9} + 82944 p^{2} T^{10} - 16320 p^{3} T^{11} + 526 p^{4} T^{12} + 384 p^{5} T^{13} - 48 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( 1 - 248 T^{2} + 29208 T^{4} - 2129944 T^{6} + 2560930 p T^{8} - 2129944 p^{2} T^{10} + 29208 p^{4} T^{12} - 248 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( 1 + 24 T^{2} - 324 T^{4} + 63336 T^{6} + 5311334 T^{8} + 63336 p^{2} T^{10} - 324 p^{4} T^{12} + 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 - 8 T - 108 T^{2} + 432 T^{3} + 10994 T^{4} - 16248 T^{5} - 716368 T^{6} + 478072 T^{7} + 33653619 T^{8} + 478072 p T^{9} - 716368 p^{2} T^{10} - 16248 p^{3} T^{11} + 10994 p^{4} T^{12} + 432 p^{5} T^{13} - 108 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 53 | \( 1 + 8 T - 60 T^{2} - 1488 T^{3} - 2854 T^{4} + 86520 T^{5} + 678032 T^{6} - 2016616 T^{7} - 44462109 T^{8} - 2016616 p T^{9} + 678032 p^{2} T^{10} + 86520 p^{3} T^{11} - 2854 p^{4} T^{12} - 1488 p^{5} T^{13} - 60 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 59 | \( 1 + 24 T + 260 T^{2} + 1968 T^{3} + 10594 T^{4} + 3240 T^{5} - 802192 T^{6} - 11964744 T^{7} - 110029565 T^{8} - 11964744 p T^{9} - 802192 p^{2} T^{10} + 3240 p^{3} T^{11} + 10594 p^{4} T^{12} + 1968 p^{5} T^{13} + 260 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( 1 - 48 T + 1248 T^{2} - 23040 T^{3} + 335856 T^{4} - 4102848 T^{5} + 43415616 T^{6} - 404880144 T^{7} + 3352897967 T^{8} - 404880144 p T^{9} + 43415616 p^{2} T^{10} - 4102848 p^{3} T^{11} + 335856 p^{4} T^{12} - 23040 p^{5} T^{13} + 1248 p^{6} T^{14} - 48 p^{7} T^{15} + p^{8} T^{16} \) |
| 67 | \( 1 + 48 T + 1292 T^{2} + 25152 T^{3} + 391386 T^{4} + 5111088 T^{5} + 57558448 T^{6} + 567562704 T^{7} + 4939433987 T^{8} + 567562704 p T^{9} + 57558448 p^{2} T^{10} + 5111088 p^{3} T^{11} + 391386 p^{4} T^{12} + 25152 p^{5} T^{13} + 1292 p^{6} T^{14} + 48 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( 1 - 208 T^{2} + 27604 T^{4} - 2815216 T^{6} + 229790374 T^{8} - 2815216 p^{2} T^{10} + 27604 p^{4} T^{12} - 208 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( 1 - 48 T + 1280 T^{2} - 24576 T^{3} + 373824 T^{4} - 4762272 T^{5} + 52785664 T^{6} - 521722704 T^{7} + 4670755679 T^{8} - 521722704 p T^{9} + 52785664 p^{2} T^{10} - 4762272 p^{3} T^{11} + 373824 p^{4} T^{12} - 24576 p^{5} T^{13} + 1280 p^{6} T^{14} - 48 p^{7} T^{15} + p^{8} T^{16} \) |
| 79 | \( 1 + 24 T + 452 T^{2} + 6240 T^{3} + 73866 T^{4} + 803640 T^{5} + 8125840 T^{6} + 79110984 T^{7} + 724752659 T^{8} + 79110984 p T^{9} + 8125840 p^{2} T^{10} + 803640 p^{3} T^{11} + 73866 p^{4} T^{12} + 6240 p^{5} T^{13} + 452 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 83 | \( ( 1 + 76 T^{2} - 192 T^{3} + 11094 T^{4} - 192 p T^{5} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 - 48 T + 1092 T^{2} - 15552 T^{3} + 140124 T^{4} - 513240 T^{5} - 6718968 T^{6} + 157471440 T^{7} - 1835542177 T^{8} + 157471440 p T^{9} - 6718968 p^{2} T^{10} - 513240 p^{3} T^{11} + 140124 p^{4} T^{12} - 15552 p^{5} T^{13} + 1092 p^{6} T^{14} - 48 p^{7} T^{15} + p^{8} T^{16} \) |
| 97 | \( 1 - 192 T^{2} + 44928 T^{4} - 5229504 T^{6} + 668134658 T^{8} - 5229504 p^{2} T^{10} + 44928 p^{4} T^{12} - 192 p^{6} T^{14} + p^{8} T^{16} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.69448858859817415844306064825, −3.68391186840654328555165093874, −3.46330762847350114706536302282, −3.34352264059934084470339719203, −3.32798851699429021435430050091, −3.06611282732966051831083467397, −2.92918950098814766413657101988, −2.82389149292745278358306307615, −2.73987183829017873231517444298, −2.73863462911641394154761906157, −2.72026143024608748588350629262, −2.36938573962216454240367386577, −2.30849322921343581710203366781, −2.14885795969099331165468221763, −1.93923127916025190827610127816, −1.57816748932413924067744591539, −1.54802351958489141110140687070, −1.39542168184524185670905075947, −1.35244767761032014593947089619, −1.04089310621657776749882009471, −0.799625656603385002360200813864, −0.67595468554384019817879051146, −0.67379351082517495316478752997, −0.58674529577124584020170950987, −0.34695404653939249825214405539,
0.34695404653939249825214405539, 0.58674529577124584020170950987, 0.67379351082517495316478752997, 0.67595468554384019817879051146, 0.799625656603385002360200813864, 1.04089310621657776749882009471, 1.35244767761032014593947089619, 1.39542168184524185670905075947, 1.54802351958489141110140687070, 1.57816748932413924067744591539, 1.93923127916025190827610127816, 2.14885795969099331165468221763, 2.30849322921343581710203366781, 2.36938573962216454240367386577, 2.72026143024608748588350629262, 2.73863462911641394154761906157, 2.73987183829017873231517444298, 2.82389149292745278358306307615, 2.92918950098814766413657101988, 3.06611282732966051831083467397, 3.32798851699429021435430050091, 3.34352264059934084470339719203, 3.46330762847350114706536302282, 3.68391186840654328555165093874, 3.69448858859817415844306064825
Plot not available for L-functions of degree greater than 10.
