| L(s) = 1 | + 15·5-s + 3·7-s − 24·11-s − 3·13-s + 60·17-s − 54·19-s + 99·23-s + 75·25-s − 54·29-s − 72·31-s + 45·35-s − 36·37-s + 411·41-s − 444·43-s − 75·47-s + 417·49-s − 342·53-s − 360·55-s + 297·59-s − 684·61-s − 45·65-s − 12·67-s − 1.28e3·71-s − 132·73-s − 72·77-s − 1.12e3·79-s + 90·83-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.161·7-s − 0.657·11-s − 0.0640·13-s + 0.856·17-s − 0.652·19-s + 0.897·23-s + 3/5·25-s − 0.345·29-s − 0.417·31-s + 0.217·35-s − 0.159·37-s + 1.56·41-s − 1.57·43-s − 0.232·47-s + 1.21·49-s − 0.886·53-s − 0.882·55-s + 0.655·59-s − 1.43·61-s − 0.0858·65-s − 0.0218·67-s − 2.14·71-s − 0.211·73-s − 0.106·77-s − 1.59·79-s + 0.119·83-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
| \(L(2)\) |
\(\approx\) |
\(0.4593032879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4593032879\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 3 T - 408 T^{2} + 4337 T^{3} + 22476 T^{4} - 659859 T^{5} + 27810882 T^{6} - 659859 p^{3} T^{7} + 22476 p^{6} T^{8} + 4337 p^{9} T^{9} - 408 p^{12} T^{10} - 3 p^{15} T^{11} + p^{18} T^{12} \) |
| 11 | \( 1 + 24 T - 1992 T^{2} - 31728 T^{3} + 186936 p T^{4} - 1672680 T^{5} - 2902945898 T^{6} - 1672680 p^{3} T^{7} + 186936 p^{7} T^{8} - 31728 p^{9} T^{9} - 1992 p^{12} T^{10} + 24 p^{15} T^{11} + p^{18} T^{12} \) |
| 13 | \( 1 + 3 T - 54 p T^{2} - 9991 T^{3} - 1074732 T^{4} + 9418107 T^{5} + 20091675672 T^{6} + 9418107 p^{3} T^{7} - 1074732 p^{6} T^{8} - 9991 p^{9} T^{9} - 54 p^{13} T^{10} + 3 p^{15} T^{11} + p^{18} T^{12} \) |
| 17 | \( ( 1 - 30 T + 8571 T^{2} - 192828 T^{3} + 8571 p^{3} T^{4} - 30 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 19 | \( ( 1 + 27 T + 10608 T^{2} + 34717 p T^{3} + 10608 p^{3} T^{4} + 27 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 23 | \( 1 - 99 T - 29352 T^{2} + 970569 T^{3} + 816670416 T^{4} - 15215127687 T^{5} - 10627423224302 T^{6} - 15215127687 p^{3} T^{7} + 816670416 p^{6} T^{8} + 970569 p^{9} T^{9} - 29352 p^{12} T^{10} - 99 p^{15} T^{11} + p^{18} T^{12} \) |
| 29 | \( 1 + 54 T - 70608 T^{2} - 1289952 T^{3} + 3470575044 T^{4} + 31834598022 T^{5} - 97644194286986 T^{6} + 31834598022 p^{3} T^{7} + 3470575044 p^{6} T^{8} - 1289952 p^{9} T^{9} - 70608 p^{12} T^{10} + 54 p^{15} T^{11} + p^{18} T^{12} \) |
| 31 | \( 1 + 72 T - 68280 T^{2} - 1517320 T^{3} + 2963351064 T^{4} - 1228509528 T^{5} - 100919502631602 T^{6} - 1228509528 p^{3} T^{7} + 2963351064 p^{6} T^{8} - 1517320 p^{9} T^{9} - 68280 p^{12} T^{10} + 72 p^{15} T^{11} + p^{18} T^{12} \) |
| 37 | \( ( 1 + 18 T + 104499 T^{2} - 1140212 T^{3} + 104499 p^{3} T^{4} + 18 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 41 | \( 1 - 411 T - 73977 T^{2} + 11118432 T^{3} + 19458541041 T^{4} - 1666084429365 T^{5} - 1136480341638098 T^{6} - 1666084429365 p^{3} T^{7} + 19458541041 p^{6} T^{8} + 11118432 p^{9} T^{9} - 73977 p^{12} T^{10} - 411 p^{15} T^{11} + p^{18} T^{12} \) |
| 43 | \( 1 + 444 T - 34605 T^{2} - 37783108 T^{3} + 4048762530 T^{4} + 2745468491196 T^{5} + 436623851528091 T^{6} + 2745468491196 p^{3} T^{7} + 4048762530 p^{6} T^{8} - 37783108 p^{9} T^{9} - 34605 p^{12} T^{10} + 444 p^{15} T^{11} + p^{18} T^{12} \) |
| 47 | \( 1 + 75 T - 174684 T^{2} - 24209925 T^{3} + 12376122924 T^{4} + 1402216717575 T^{5} - 841849814761454 T^{6} + 1402216717575 p^{3} T^{7} + 12376122924 p^{6} T^{8} - 24209925 p^{9} T^{9} - 174684 p^{12} T^{10} + 75 p^{15} T^{11} + p^{18} T^{12} \) |
| 53 | \( ( 1 + 171 T + 455763 T^{2} + 51067422 T^{3} + 455763 p^{3} T^{4} + 171 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 59 | \( 1 - 297 T - 29919 T^{2} - 188413830 T^{3} + 12462691251 T^{4} + 10939693886847 T^{5} + 16616892260928022 T^{6} + 10939693886847 p^{3} T^{7} + 12462691251 p^{6} T^{8} - 188413830 p^{9} T^{9} - 29919 p^{12} T^{10} - 297 p^{15} T^{11} + p^{18} T^{12} \) |
| 61 | \( 1 + 684 T - 63027 T^{2} - 265320364 T^{3} - 53591557446 T^{4} + 610010798364 p T^{5} + 34048157230279581 T^{6} + 610010798364 p^{4} T^{7} - 53591557446 p^{6} T^{8} - 265320364 p^{9} T^{9} - 63027 p^{12} T^{10} + 684 p^{15} T^{11} + p^{18} T^{12} \) |
| 67 | \( 1 + 12 T - 698493 T^{2} - 53055508 T^{3} + 277600862826 T^{4} + 17504353163196 T^{5} - 91456777656400413 T^{6} + 17504353163196 p^{3} T^{7} + 277600862826 p^{6} T^{8} - 53055508 p^{9} T^{9} - 698493 p^{12} T^{10} + 12 p^{15} T^{11} + p^{18} T^{12} \) |
| 71 | \( ( 1 + 642 T + 197304 T^{2} - 87694746 T^{3} + 197304 p^{3} T^{4} + 642 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( ( 1 + 66 T + 335715 T^{2} - 146251676 T^{3} + 335715 p^{3} T^{4} + 66 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 79 | \( 1 + 1122 T - 428073 T^{2} - 266737918 T^{3} + 802606767534 T^{4} + 227824373924802 T^{5} - 303553868663922141 T^{6} + 227824373924802 p^{3} T^{7} + 802606767534 p^{6} T^{8} - 266737918 p^{9} T^{9} - 428073 p^{12} T^{10} + 1122 p^{15} T^{11} + p^{18} T^{12} \) |
| 83 | \( 1 - 90 T - 1227801 T^{2} + 6466230 T^{3} + 815482777314 T^{4} + 301749773130 p T^{5} - 72388057449929 p^{2} T^{6} + 301749773130 p^{4} T^{7} + 815482777314 p^{6} T^{8} + 6466230 p^{9} T^{9} - 1227801 p^{12} T^{10} - 90 p^{15} T^{11} + p^{18} T^{12} \) |
| 89 | \( ( 1 + 756 T + 1118256 T^{2} + 284981598 T^{3} + 1118256 p^{3} T^{4} + 756 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 97 | \( 1 + 1536 T - 673023 T^{2} - 878940928 T^{3} + 2175347162658 T^{4} + 1171979614210944 T^{5} - 1173318191949724767 T^{6} + 1171979614210944 p^{3} T^{7} + 2175347162658 p^{6} T^{8} - 878940928 p^{9} T^{9} - 673023 p^{12} T^{10} + 1536 p^{15} T^{11} + p^{18} T^{12} \) |
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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
−4.47670032312151331369889042817, −4.41164904408675548923730292285, −4.32709546181519320358296256189, −4.30104672541074812795919344316, −3.77312880881898991305815714749, −3.56942290534773526862045637260, −3.51386121595884232228468648708, −3.46173823967253549778099546277, −3.25248615400519986050919825075, −3.21218062919657713522452473773, −2.76001293225860158413120714737, −2.52612500074183597435230067988, −2.48301331010810289546389536707, −2.41134705313927625975859043528, −2.32889423483688059881804612852, −2.21406537900170011577736987205, −1.62695826077331909101729137885, −1.56793379358235794859788947560, −1.31377123973333556179494423062, −1.28446791093291406344876863147, −1.12463376127378776724957614208, −1.11651925315463418446353993631, −0.47387066778790023953210737776, −0.15475499907195436424043034037, −0.10488064277868972119346716566,
0.10488064277868972119346716566, 0.15475499907195436424043034037, 0.47387066778790023953210737776, 1.11651925315463418446353993631, 1.12463376127378776724957614208, 1.28446791093291406344876863147, 1.31377123973333556179494423062, 1.56793379358235794859788947560, 1.62695826077331909101729137885, 2.21406537900170011577736987205, 2.32889423483688059881804612852, 2.41134705313927625975859043528, 2.48301331010810289546389536707, 2.52612500074183597435230067988, 2.76001293225860158413120714737, 3.21218062919657713522452473773, 3.25248615400519986050919825075, 3.46173823967253549778099546277, 3.51386121595884232228468648708, 3.56942290534773526862045637260, 3.77312880881898991305815714749, 4.30104672541074812795919344316, 4.32709546181519320358296256189, 4.41164904408675548923730292285, 4.47670032312151331369889042817
Plot not available for L-functions of degree greater than 10.
