L(s) = 1 | + 9·3-s + 4·5-s − 10·7-s + 45·9-s + 20·11-s + 21·13-s + 36·15-s − 2·17-s + 10·19-s − 90·21-s + 2·23-s + 43·25-s + 162·27-s + 114·29-s + 180·33-s − 40·35-s + 189·39-s + 48·41-s + 21·43-s + 180·45-s − 46·47-s − 217·49-s − 18·51-s − 18·53-s + 80·55-s + 90·57-s + 144·59-s + ⋯ |
L(s) = 1 | + 3·3-s + 4/5·5-s − 1.42·7-s + 5·9-s + 1.81·11-s + 1.61·13-s + 12/5·15-s − 0.117·17-s + 0.526·19-s − 4.28·21-s + 2/23·23-s + 1.71·25-s + 6·27-s + 3.93·29-s + 5.45·33-s − 8/7·35-s + 4.84·39-s + 1.17·41-s + 0.488·43-s + 4·45-s − 0.978·47-s − 4.42·49-s − 0.352·51-s − 0.339·53-s + 1.45·55-s + 1.57·57-s + 2.44·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(29.28068240\) |
\(L(\frac12)\) |
\(\approx\) |
\(29.28068240\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( ( 1 - p T + p T^{2} )^{3} \) |
| 19 | \( 1 - 10 T + 29 p T^{2} - 28 p^{2} T^{3} + 29 p^{3} T^{4} - 10 p^{4} T^{5} + p^{6} T^{6} \) |
good | 5 | \( 1 - 4 T - 27 T^{2} + 12 p^{2} T^{3} - 34 p T^{4} - 4048 T^{5} + 30721 T^{6} - 4048 p^{2} T^{7} - 34 p^{5} T^{8} + 12 p^{8} T^{9} - 27 p^{8} T^{10} - 4 p^{10} T^{11} + p^{12} T^{12} \) |
| 7 | \( ( 1 + 5 T + 146 T^{2} + 473 T^{3} + 146 p^{2} T^{4} + 5 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 11 | \( ( 1 - 10 T + 163 T^{2} - 920 T^{3} + 163 p^{2} T^{4} - 10 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 13 | \( 1 - 21 T + 567 T^{2} - 8820 T^{3} + 151893 T^{4} - 2188599 T^{5} + 30651622 T^{6} - 2188599 p^{2} T^{7} + 151893 p^{4} T^{8} - 8820 p^{6} T^{9} + 567 p^{8} T^{10} - 21 p^{10} T^{11} + p^{12} T^{12} \) |
| 17 | \( 1 + 2 T - 195 T^{2} - 8490 T^{3} - 26798 T^{4} + 821354 T^{5} + 56245501 T^{6} + 821354 p^{2} T^{7} - 26798 p^{4} T^{8} - 8490 p^{6} T^{9} - 195 p^{8} T^{10} + 2 p^{10} T^{11} + p^{12} T^{12} \) |
| 23 | \( 1 - 2 T - 651 T^{2} + 22530 T^{3} + 58690 T^{4} - 6981302 T^{5} + 210084589 T^{6} - 6981302 p^{2} T^{7} + 58690 p^{4} T^{8} + 22530 p^{6} T^{9} - 651 p^{8} T^{10} - 2 p^{10} T^{11} + p^{12} T^{12} \) |
| 29 | \( 1 - 114 T + 6651 T^{2} - 264366 T^{3} + 7406082 T^{4} - 162169266 T^{5} + 3923139067 T^{6} - 162169266 p^{2} T^{7} + 7406082 p^{4} T^{8} - 264366 p^{6} T^{9} + 6651 p^{8} T^{10} - 114 p^{10} T^{11} + p^{12} T^{12} \) |
| 31 | \( 1 - 2589 T^{2} + 4126398 T^{4} - 4902421457 T^{6} + 4126398 p^{4} T^{8} - 2589 p^{8} T^{10} + p^{12} T^{12} \) |
| 37 | \( 1 - 7101 T^{2} + 22283478 T^{4} - 39526739993 T^{6} + 22283478 p^{4} T^{8} - 7101 p^{8} T^{10} + p^{12} T^{12} \) |
| 41 | \( 1 - 48 T + 3267 T^{2} - 119952 T^{3} + 4805910 T^{4} - 295710864 T^{5} + 8758627639 T^{6} - 295710864 p^{2} T^{7} + 4805910 p^{4} T^{8} - 119952 p^{6} T^{9} + 3267 p^{8} T^{10} - 48 p^{10} T^{11} + p^{12} T^{12} \) |
| 43 | \( 1 - 21 T - 2589 T^{2} + 44908 T^{3} + 2438781 T^{4} - 926151 T^{5} - 2895324882 T^{6} - 926151 p^{2} T^{7} + 2438781 p^{4} T^{8} + 44908 p^{6} T^{9} - 2589 p^{8} T^{10} - 21 p^{10} T^{11} + p^{12} T^{12} \) |
| 47 | \( 1 + 46 T - 3627 T^{2} - 105366 T^{3} + 11796250 T^{4} + 118472518 T^{5} - 27483628595 T^{6} + 118472518 p^{2} T^{7} + 11796250 p^{4} T^{8} - 105366 p^{6} T^{9} - 3627 p^{8} T^{10} + 46 p^{10} T^{11} + p^{12} T^{12} \) |
| 53 | \( 1 + 18 T + 1983 T^{2} + 33750 T^{3} - 3296514 T^{4} - 624923658 T^{5} - 13195866269 T^{6} - 624923658 p^{2} T^{7} - 3296514 p^{4} T^{8} + 33750 p^{6} T^{9} + 1983 p^{8} T^{10} + 18 p^{10} T^{11} + p^{12} T^{12} \) |
| 59 | \( 1 - 144 T + 18363 T^{2} - 1648944 T^{3} + 136757526 T^{4} - 9108037188 T^{5} + 587755175695 T^{6} - 9108037188 p^{2} T^{7} + 136757526 p^{4} T^{8} - 1648944 p^{6} T^{9} + 18363 p^{8} T^{10} - 144 p^{10} T^{11} + p^{12} T^{12} \) |
| 61 | \( 1 - 19 T - 8341 T^{2} + 58452 T^{3} + 41218657 T^{4} - 6335873 T^{5} - 171209104538 T^{6} - 6335873 p^{2} T^{7} + 41218657 p^{4} T^{8} + 58452 p^{6} T^{9} - 8341 p^{8} T^{10} - 19 p^{10} T^{11} + p^{12} T^{12} \) |
| 67 | \( 1 + 3 p T + 17067 T^{2} + 10800 p T^{3} - 15078495 T^{4} - 4688601321 T^{5} - 383637813770 T^{6} - 4688601321 p^{2} T^{7} - 15078495 p^{4} T^{8} + 10800 p^{7} T^{9} + 17067 p^{8} T^{10} + 3 p^{11} T^{11} + p^{12} T^{12} \) |
| 71 | \( 1 + 204 T + 32355 T^{2} + 3770532 T^{3} + 391625478 T^{4} + 33732786828 T^{5} + 2591173356487 T^{6} + 33732786828 p^{2} T^{7} + 391625478 p^{4} T^{8} + 3770532 p^{6} T^{9} + 32355 p^{8} T^{10} + 204 p^{10} T^{11} + p^{12} T^{12} \) |
| 73 | \( 1 - 51 T - 13677 T^{2} + 236684 T^{3} + 151022841 T^{4} - 1283462985 T^{5} - 898689370938 T^{6} - 1283462985 p^{2} T^{7} + 151022841 p^{4} T^{8} + 236684 p^{6} T^{9} - 13677 p^{8} T^{10} - 51 p^{10} T^{11} + p^{12} T^{12} \) |
| 79 | \( 1 + 153 T + 18543 T^{2} + 1643220 T^{3} + 91835673 T^{4} + 3204682227 T^{5} + 172467894238 T^{6} + 3204682227 p^{2} T^{7} + 91835673 p^{4} T^{8} + 1643220 p^{6} T^{9} + 18543 p^{8} T^{10} + 153 p^{10} T^{11} + p^{12} T^{12} \) |
| 83 | \( ( 1 + 26 T + 18127 T^{2} + 338860 T^{3} + 18127 p^{2} T^{4} + 26 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 89 | \( 1 - 216 T + 42231 T^{2} - 5762664 T^{3} + 746889450 T^{4} - 77765731020 T^{5} + 7573479675919 T^{6} - 77765731020 p^{2} T^{7} + 746889450 p^{4} T^{8} - 5762664 p^{6} T^{9} + 42231 p^{8} T^{10} - 216 p^{10} T^{11} + p^{12} T^{12} \) |
| 97 | \( 1 - 12 T + 12579 T^{2} - 150372 T^{3} + 36107670 T^{4} + 10504810740 T^{5} - 3467999369 T^{6} + 10504810740 p^{2} T^{7} + 36107670 p^{4} T^{8} - 150372 p^{6} T^{9} + 12579 p^{8} T^{10} - 12 p^{10} T^{11} + p^{12} T^{12} \) |
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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
−4.98572244283426308855055617550, −4.76810193124437887736241164596, −4.69709654402857159058453353038, −4.54261027801204382323201679646, −4.38997411967131505076172832092, −4.34291956963371516627656619976, −4.03005588386186753336879434595, −3.93674736618948563618262449792, −3.53962368022156263848715364973, −3.37910655365723759015663699629, −3.33014557193805627689758685512, −3.25301565100012479931877187388, −2.98857801430246056974404006898, −2.98434340768286780484732593545, −2.75906397951420361617634491683, −2.69769494139425886567791346450, −2.21733805889728692996327631356, −1.99901358330509595590141519050, −1.80893131289660265760507515113, −1.74132362186798608718608667637, −1.35344157380521844232276869654, −1.05870693406240181496591700139, −0.957781701483161854104454035919, −0.925378081206644798170276309720, −0.24123944636559081948350895661,
0.24123944636559081948350895661, 0.925378081206644798170276309720, 0.957781701483161854104454035919, 1.05870693406240181496591700139, 1.35344157380521844232276869654, 1.74132362186798608718608667637, 1.80893131289660265760507515113, 1.99901358330509595590141519050, 2.21733805889728692996327631356, 2.69769494139425886567791346450, 2.75906397951420361617634491683, 2.98434340768286780484732593545, 2.98857801430246056974404006898, 3.25301565100012479931877187388, 3.33014557193805627689758685512, 3.37910655365723759015663699629, 3.53962368022156263848715364973, 3.93674736618948563618262449792, 4.03005588386186753336879434595, 4.34291956963371516627656619976, 4.38997411967131505076172832092, 4.54261027801204382323201679646, 4.69709654402857159058453353038, 4.76810193124437887736241164596, 4.98572244283426308855055617550
Plot not available for L-functions of degree greater than 10.