| L(s) = 1 | + 42·7-s − 27·9-s − 72·17-s − 196·23-s + 498·25-s + 104·31-s − 216·41-s − 408·47-s + 1.02e3·49-s − 1.13e3·63-s − 1.16e3·71-s − 276·73-s + 1.35e3·79-s + 486·81-s + 2.52e3·89-s − 2.34e3·97-s + 2.78e3·103-s − 2.58e3·113-s − 3.02e3·119-s + 3.27e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 1.94e3·153-s + ⋯ |
| L(s) = 1 | + 2.26·7-s − 9-s − 1.02·17-s − 1.77·23-s + 3.98·25-s + 0.602·31-s − 0.822·41-s − 1.26·47-s + 3·49-s − 2.26·63-s − 1.94·71-s − 0.442·73-s + 1.92·79-s + 2/3·81-s + 3.00·89-s − 2.44·97-s + 2.66·103-s − 2.14·113-s − 2.32·119-s + 2.45·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 1.02·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(15.85112178\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.85112178\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p^{2} T^{2} )^{3} \) |
| 7 | \( ( 1 - p T )^{6} \) |
| good | 5 | \( 1 - 498 T^{2} + 120027 T^{4} - 18269844 T^{6} + 120027 p^{6} T^{8} - 498 p^{12} T^{10} + p^{18} T^{12} \) |
| 11 | \( 1 - 3270 T^{2} + 8076603 T^{4} - 12268232412 T^{6} + 8076603 p^{6} T^{8} - 3270 p^{12} T^{10} + p^{18} T^{12} \) |
| 13 | \( 1 - 6298 T^{2} + 13800455 T^{4} - 20942216428 T^{6} + 13800455 p^{6} T^{8} - 6298 p^{12} T^{10} + p^{18} T^{12} \) |
| 17 | \( ( 1 + 36 T + 3413 T^{2} + 694188 T^{3} + 3413 p^{3} T^{4} + 36 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 19 | \( 1 - 206 p T^{2} + 107507975 T^{4} - 346702513676 T^{6} + 107507975 p^{6} T^{8} - 206 p^{13} T^{10} + p^{18} T^{12} \) |
| 23 | \( ( 1 + 98 T + 32235 T^{2} + 1929116 T^{3} + 32235 p^{3} T^{4} + 98 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 29 | \( 1 + 32186 T^{2} + 1405063335 T^{4} + 38022140878636 T^{6} + 1405063335 p^{6} T^{8} + 32186 p^{12} T^{10} + p^{18} T^{12} \) |
| 31 | \( ( 1 - 52 T + 29105 T^{2} + 3528392 T^{3} + 29105 p^{3} T^{4} - 52 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 37 | \( 1 - 31254 T^{2} + 1099258743 T^{4} - 76789777860340 T^{6} + 1099258743 p^{6} T^{8} - 31254 p^{12} T^{10} + p^{18} T^{12} \) |
| 41 | \( ( 1 + 108 T + 198893 T^{2} + 14030484 T^{3} + 198893 p^{3} T^{4} + 108 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 43 | \( 1 - 211270 T^{2} + 27158501447 T^{4} - 2374881802425460 T^{6} + 27158501447 p^{6} T^{8} - 211270 p^{12} T^{10} + p^{18} T^{12} \) |
| 47 | \( ( 1 + 204 T + 38141 T^{2} - 36096600 T^{3} + 38141 p^{3} T^{4} + 204 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 53 | \( 1 - 293862 T^{2} + 25816491735 T^{4} - 255200679487060 T^{6} + 25816491735 p^{6} T^{8} - 293862 p^{12} T^{10} + p^{18} T^{12} \) |
| 59 | \( 1 - 121986 T^{2} + 29011638855 T^{4} - 18729620552449980 T^{6} + 29011638855 p^{6} T^{8} - 121986 p^{12} T^{10} + p^{18} T^{12} \) |
| 61 | \( 1 - 611418 T^{2} + 227998482855 T^{4} - 63536980209731692 T^{6} + 227998482855 p^{6} T^{8} - 611418 p^{12} T^{10} + p^{18} T^{12} \) |
| 67 | \( 1 - 388998 T^{2} + 93671348823 T^{4} - 1045103396994484 T^{6} + 93671348823 p^{6} T^{8} - 388998 p^{12} T^{10} + p^{18} T^{12} \) |
| 71 | \( ( 1 + 582 T + 935219 T^{2} + 326943684 T^{3} + 935219 p^{3} T^{4} + 582 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( ( 1 + 138 T + 196095 T^{2} - 293744388 T^{3} + 196095 p^{3} T^{4} + 138 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 79 | \( ( 1 - 676 T + 964933 T^{2} - 319327160 T^{3} + 964933 p^{3} T^{4} - 676 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 83 | \( 1 - 937250 T^{2} + 840335913175 T^{4} - 485522015972160700 T^{6} + 840335913175 p^{6} T^{8} - 937250 p^{12} T^{10} + p^{18} T^{12} \) |
| 89 | \( ( 1 - 1260 T + 2067965 T^{2} - 1773208500 T^{3} + 2067965 p^{3} T^{4} - 1260 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 97 | \( ( 1 + 1170 T + 2333943 T^{2} + 1862082124 T^{3} + 2333943 p^{3} T^{4} + 1170 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.67147419853560881523051873498, −4.58180925487412965996083935842, −4.33982936134228476261603737008, −4.10753427219545567050998479413, −4.08284979192581246403413044012, −4.06819452427401389676801926169, −3.85276165867539497568737098049, −3.24289486135001136117661048540, −3.18079324723949205046635477838, −3.09998932818496412143555074587, −3.01025554017819658725725653248, −2.94168915087885847971282409645, −2.88200096537152070158847244779, −2.18870158243757033794223627887, −2.15290106472040428084577248350, −2.00859182950842170350448374897, −1.86182834043154434234841811217, −1.84640504921589164662751835853, −1.69394590256010139669944487414, −1.04150475599401310023662142440, −1.00439410272915707005584888654, −0.973843459359573591020063951289, −0.55952634024298327087978928516, −0.46134563883351881111294679726, −0.27028106391416598070238583376,
0.27028106391416598070238583376, 0.46134563883351881111294679726, 0.55952634024298327087978928516, 0.973843459359573591020063951289, 1.00439410272915707005584888654, 1.04150475599401310023662142440, 1.69394590256010139669944487414, 1.84640504921589164662751835853, 1.86182834043154434234841811217, 2.00859182950842170350448374897, 2.15290106472040428084577248350, 2.18870158243757033794223627887, 2.88200096537152070158847244779, 2.94168915087885847971282409645, 3.01025554017819658725725653248, 3.09998932818496412143555074587, 3.18079324723949205046635477838, 3.24289486135001136117661048540, 3.85276165867539497568737098049, 4.06819452427401389676801926169, 4.08284979192581246403413044012, 4.10753427219545567050998479413, 4.33982936134228476261603737008, 4.58180925487412965996083935842, 4.67147419853560881523051873498
Plot not available for L-functions of degree greater than 10.
