| L(s) = 1 | + 2-s + 4-s − 15·5-s − 44·7-s − 18·8-s − 15·10-s − 38·11-s − 28·13-s − 44·14-s − 20·16-s − 38·17-s + 374·19-s − 15·20-s − 38·22-s + 81·23-s + 75·25-s − 28·26-s − 44·28-s − 160·29-s − 227·31-s + 36·32-s − 38·34-s + 660·35-s + 156·37-s + 374·38-s + 270·40-s + 338·41-s + ⋯ |
| L(s) = 1 | + 0.353·2-s + 1/8·4-s − 1.34·5-s − 2.37·7-s − 0.795·8-s − 0.474·10-s − 1.04·11-s − 0.597·13-s − 0.839·14-s − 0.312·16-s − 0.542·17-s + 4.51·19-s − 0.167·20-s − 0.368·22-s + 0.734·23-s + 3/5·25-s − 0.211·26-s − 0.296·28-s − 1.02·29-s − 1.31·31-s + 0.198·32-s − 0.191·34-s + 3.18·35-s + 0.693·37-s + 1.59·38-s + 1.06·40-s + 1.28·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.275827698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.275827698\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 + p T + p^{2} T^{2} )^{3} \) |
| good | 2 | \( 1 - T + 19 T^{3} - 17 T^{4} - 29 p T^{5} + 263 p^{2} T^{6} - 29 p^{4} T^{7} - 17 p^{6} T^{8} + 19 p^{9} T^{9} - p^{15} T^{11} + p^{18} T^{12} \) |
| 7 | \( 1 + 44 T + 355 T^{2} + 5596 T^{3} + 549982 T^{4} + 8022356 T^{5} + 18565583 T^{6} + 8022356 p^{3} T^{7} + 549982 p^{6} T^{8} + 5596 p^{9} T^{9} + 355 p^{12} T^{10} + 44 p^{15} T^{11} + p^{18} T^{12} \) |
| 11 | \( 1 + 38 T + 63 T^{2} + 16726 T^{3} - 610238 T^{4} - 47690362 T^{5} - 61118617 T^{6} - 47690362 p^{3} T^{7} - 610238 p^{6} T^{8} + 16726 p^{9} T^{9} + 63 p^{12} T^{10} + 38 p^{15} T^{11} + p^{18} T^{12} \) |
| 13 | \( 1 + 28 T - 1231 T^{2} + 10596 T^{3} - 1008266 T^{4} - 5122396 p T^{5} + 9029425909 T^{6} - 5122396 p^{4} T^{7} - 1008266 p^{6} T^{8} + 10596 p^{9} T^{9} - 1231 p^{12} T^{10} + 28 p^{15} T^{11} + p^{18} T^{12} \) |
| 17 | \( ( 1 + 19 T + 3262 T^{2} - 367193 T^{3} + 3262 p^{3} T^{4} + 19 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 19 | \( ( 1 - 187 T + 24164 T^{2} - 2039395 T^{3} + 24164 p^{3} T^{4} - 187 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 23 | \( 1 - 81 T - 6639 T^{2} - 1215054 T^{3} + 19542687 T^{4} + 14989642191 T^{5} + 516382122526 T^{6} + 14989642191 p^{3} T^{7} + 19542687 p^{6} T^{8} - 1215054 p^{9} T^{9} - 6639 p^{12} T^{10} - 81 p^{15} T^{11} + p^{18} T^{12} \) |
| 29 | \( 1 + 160 T + 201 T^{2} + 4240400 T^{3} + 39777790 T^{4} - 105296479880 T^{5} - 1938960010651 T^{6} - 105296479880 p^{3} T^{7} + 39777790 p^{6} T^{8} + 4240400 p^{9} T^{9} + 201 p^{12} T^{10} + 160 p^{15} T^{11} + p^{18} T^{12} \) |
| 31 | \( 1 + 227 T - 641 p T^{2} - 11335070 T^{3} - 155233145 T^{4} + 219050654987 T^{5} + 44926162742318 T^{6} + 219050654987 p^{3} T^{7} - 155233145 p^{6} T^{8} - 11335070 p^{9} T^{9} - 641 p^{13} T^{10} + 227 p^{15} T^{11} + p^{18} T^{12} \) |
| 37 | \( ( 1 - 78 T + 52035 T^{2} + 5735212 T^{3} + 52035 p^{3} T^{4} - 78 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 41 | \( 1 - 338 T - 49707 T^{2} + 13532474 T^{3} + 3928066402 T^{4} + 381000011182 T^{5} - 552045080998267 T^{6} + 381000011182 p^{3} T^{7} + 3928066402 p^{6} T^{8} + 13532474 p^{9} T^{9} - 49707 p^{12} T^{10} - 338 p^{15} T^{11} + p^{18} T^{12} \) |
| 43 | \( 1 + 22 T - 78121 T^{2} + 31661814 T^{3} + 258355834 T^{4} - 1312336751722 T^{5} + 734860822875799 T^{6} - 1312336751722 p^{3} T^{7} + 258355834 p^{6} T^{8} + 31661814 p^{9} T^{9} - 78121 p^{12} T^{10} + 22 p^{15} T^{11} + p^{18} T^{12} \) |
| 47 | \( 1 - 472 T - 142893 T^{2} + 22851880 T^{3} + 49627449094 T^{4} - 4984229908888 T^{5} - 4496038100665225 T^{6} - 4984229908888 p^{3} T^{7} + 49627449094 p^{6} T^{8} + 22851880 p^{9} T^{9} - 142893 p^{12} T^{10} - 472 p^{15} T^{11} + p^{18} T^{12} \) |
| 53 | \( ( 1 - 521 T + 508018 T^{2} - 154190045 T^{3} + 508018 p^{3} T^{4} - 521 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 59 | \( 1 + 140 T - 430089 T^{2} + 16207180 T^{3} + 106591069990 T^{4} - 9443276399980 T^{5} - 24486154809286381 T^{6} - 9443276399980 p^{3} T^{7} + 106591069990 p^{6} T^{8} + 16207180 p^{9} T^{9} - 430089 p^{12} T^{10} + 140 p^{15} T^{11} + p^{18} T^{12} \) |
| 61 | \( 1 + 595 T - 324169 T^{2} - 133124136 T^{3} + 146356703437 T^{4} + 29445669373037 T^{5} - 27252272251928570 T^{6} + 29445669373037 p^{3} T^{7} + 146356703437 p^{6} T^{8} - 133124136 p^{9} T^{9} - 324169 p^{12} T^{10} + 595 p^{15} T^{11} + p^{18} T^{12} \) |
| 67 | \( 1 + 878 T - 207353 T^{2} - 174797570 T^{3} + 208940374354 T^{4} + 68373272629382 T^{5} - 36005713948020865 T^{6} + 68373272629382 p^{3} T^{7} + 208940374354 p^{6} T^{8} - 174797570 p^{9} T^{9} - 207353 p^{12} T^{10} + 878 p^{15} T^{11} + p^{18} T^{12} \) |
| 71 | \( ( 1 + 602 T + 490081 T^{2} + 150373964 T^{3} + 490081 p^{3} T^{4} + 602 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( ( 1 - 1294 T + 1071143 T^{2} - 602684716 T^{3} + 1071143 p^{3} T^{4} - 1294 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 79 | \( 1 + 629 T - 1180535 T^{2} - 253092026 T^{3} + 1315817177527 T^{4} + 156070600864037 T^{5} - 694144886678428162 T^{6} + 156070600864037 p^{3} T^{7} + 1315817177527 p^{6} T^{8} - 253092026 p^{9} T^{9} - 1180535 p^{12} T^{10} + 629 p^{15} T^{11} + p^{18} T^{12} \) |
| 83 | \( 1 - 1287 T + 306885 T^{2} + 513964746 T^{3} - 398868387501 T^{4} + 46694949648021 T^{5} + 57530891765281102 T^{6} + 46694949648021 p^{3} T^{7} - 398868387501 p^{6} T^{8} + 513964746 p^{9} T^{9} + 306885 p^{12} T^{10} - 1287 p^{15} T^{11} + p^{18} T^{12} \) |
| 89 | \( ( 1 - 2154 T + 3172479 T^{2} - 3111332052 T^{3} + 3172479 p^{3} T^{4} - 2154 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 97 | \( 1 + 1392 T - 1344291 T^{2} - 640472944 T^{3} + 4150444710726 T^{4} + 1368214665702768 T^{5} - 3140866054509007143 T^{6} + 1368214665702768 p^{3} T^{7} + 4150444710726 p^{6} T^{8} - 640472944 p^{9} T^{9} - 1344291 p^{12} T^{10} + 1392 p^{15} T^{11} + p^{18} T^{12} \) |
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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
−5.64688845676059111291399017992, −5.34793335035644911759527271075, −5.22898878675605127457248327251, −5.20940107420964925482208548350, −4.93888484382848633774736872706, −4.76057891308172857997710867637, −4.51179607368634170085823196407, −4.35194235417248260556491268150, −3.83646226465715178459108914303, −3.79127065343419629751175203851, −3.76166591245201679644262890966, −3.39375058236226087965472242223, −3.34377090671405162206838438883, −3.23966125061973645538537474728, −3.04438348941421696905258326635, −2.60821841377420355361848665918, −2.46789062674345609632864465558, −2.34458105192506310838462730947, −2.30669789818714182908466854968, −1.62474088093926047367559464018, −1.04424829941113261563497570213, −0.961002784397783907802598226639, −0.853898012077550467950396655539, −0.35138344149996091281616852386, −0.20917988851622874198767539088,
0.20917988851622874198767539088, 0.35138344149996091281616852386, 0.853898012077550467950396655539, 0.961002784397783907802598226639, 1.04424829941113261563497570213, 1.62474088093926047367559464018, 2.30669789818714182908466854968, 2.34458105192506310838462730947, 2.46789062674345609632864465558, 2.60821841377420355361848665918, 3.04438348941421696905258326635, 3.23966125061973645538537474728, 3.34377090671405162206838438883, 3.39375058236226087965472242223, 3.76166591245201679644262890966, 3.79127065343419629751175203851, 3.83646226465715178459108914303, 4.35194235417248260556491268150, 4.51179607368634170085823196407, 4.76057891308172857997710867637, 4.93888484382848633774736872706, 5.20940107420964925482208548350, 5.22898878675605127457248327251, 5.34793335035644911759527271075, 5.64688845676059111291399017992
Plot not available for L-functions of degree greater than 10.
