| L(s) = 1 | + 18·5-s + 27·9-s − 6·17-s + 87·25-s + 114·37-s + 486·45-s + 18·53-s + 318·61-s + 342·73-s + 411·81-s − 108·85-s + 150·89-s + 498·101-s + 318·109-s − 672·113-s + 363·121-s − 454·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 162·153-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 18/5·5-s + 3·9-s − 0.352·17-s + 3.47·25-s + 3.08·37-s + 54/5·45-s + 0.339·53-s + 5.21·61-s + 4.68·73-s + 5.07·81-s − 1.27·85-s + 1.68·89-s + 4.93·101-s + 2.91·109-s − 5.94·113-s + 3·121-s − 3.63·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s − 1.05·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(35.05501335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(35.05501335\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - p^{3} T^{2} + 106 p T^{4} - 2807 T^{6} + 106 p^{5} T^{8} - p^{11} T^{10} + p^{12} T^{12} \) |
| 5 | \( ( 1 - 9 T + 78 T^{2} - 421 T^{3} + 78 p^{2} T^{4} - 9 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 11 | \( 1 - 3 p^{2} T^{2} + 75918 T^{4} - 10832983 T^{6} + 75918 p^{4} T^{8} - 3 p^{10} T^{10} + p^{12} T^{12} \) |
| 13 | \( ( 1 + 339 T^{2} + 784 T^{3} + 339 p^{2} T^{4} + p^{6} T^{6} )^{2} \) |
| 17 | \( ( 1 + 3 T + 270 T^{2} + 3135 T^{3} + 270 p^{2} T^{4} + 3 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 19 | \( 1 - 411 T^{2} + 297582 T^{4} - 105762503 T^{6} + 297582 p^{4} T^{8} - 411 p^{8} T^{10} + p^{12} T^{12} \) |
| 23 | \( 1 - 579 T^{2} + 272382 T^{4} - 96432847 T^{6} + 272382 p^{4} T^{8} - 579 p^{8} T^{10} + p^{12} T^{12} \) |
| 29 | \( ( 1 + 1347 T^{2} - 5488 T^{3} + 1347 p^{2} T^{4} + p^{6} T^{6} )^{2} \) |
| 31 | \( 1 - 5235 T^{2} + 11871534 T^{4} - 14920945711 T^{6} + 11871534 p^{4} T^{8} - 5235 p^{8} T^{10} + p^{12} T^{12} \) |
| 37 | \( ( 1 - 57 T + 4518 T^{2} - 156429 T^{3} + 4518 p^{2} T^{4} - 57 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 41 | \( ( 1 + 3195 T^{2} - 22736 T^{3} + 3195 p^{2} T^{4} + p^{6} T^{6} )^{2} \) |
| 43 | \( 1 - 5718 T^{2} + 18746463 T^{4} - 40352107060 T^{6} + 18746463 p^{4} T^{8} - 5718 p^{8} T^{10} + p^{12} T^{12} \) |
| 47 | \( 1 - 8307 T^{2} + 34581678 T^{4} - 93188461807 T^{6} + 34581678 p^{4} T^{8} - 8307 p^{8} T^{10} + p^{12} T^{12} \) |
| 53 | \( ( 1 - 9 T + 3078 T^{2} - 72093 T^{3} + 3078 p^{2} T^{4} - 9 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 59 | \( 1 - 1707 T^{2} + 12042558 T^{4} - 66367351015 T^{6} + 12042558 p^{4} T^{8} - 1707 p^{8} T^{10} + p^{12} T^{12} \) |
| 61 | \( ( 1 - 159 T + 17382 T^{2} - 1180955 T^{3} + 17382 p^{2} T^{4} - 159 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 67 | \( 1 - 23547 T^{2} + 242399454 T^{4} - 1408178101687 T^{6} + 242399454 p^{4} T^{8} - 23547 p^{8} T^{10} + p^{12} T^{12} \) |
| 71 | \( 1 - 10086 T^{2} + 56255151 T^{4} - 272956768468 T^{6} + 56255151 p^{4} T^{8} - 10086 p^{8} T^{10} + p^{12} T^{12} \) |
| 73 | \( ( 1 - 171 T + 342 p T^{2} - 1956639 T^{3} + 342 p^{3} T^{4} - 171 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 79 | \( 1 - 16371 T^{2} + 146441790 T^{4} - 955309016095 T^{6} + 146441790 p^{4} T^{8} - 16371 p^{8} T^{10} + p^{12} T^{12} \) |
| 83 | \( 1 - 29238 T^{2} + 4861989 p T^{4} - 3424551164404 T^{6} + 4861989 p^{5} T^{8} - 29238 p^{8} T^{10} + p^{12} T^{12} \) |
| 89 | \( ( 1 - 75 T + 150 p T^{2} - 831039 T^{3} + 150 p^{3} T^{4} - 75 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 97 | \( ( 1 + 28059 T^{2} + 784 T^{3} + 28059 p^{2} T^{4} + p^{6} T^{6} )^{2} \) |
| 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
−4.92235331727106541157958683393, −4.50520459015240422442881368514, −4.36818163070755102673846689061, −4.35705263875738290081507153844, −4.00475554551239750582111179535, −4.00433357856061170614471890027, −3.95583534692726402967389595060, −3.68401546836073886454470142959, −3.34121664186695758072185403195, −3.33731053959229610234761154586, −3.25481103250109113844278881707, −2.81754000355144506521211399300, −2.51090266755196304743512274390, −2.20546741915855906334739078503, −2.16937215793739853827687520722, −2.13405825790575222701300227909, −2.09952992063649657576148441267, −2.08782257801348456766684296828, −1.82695044901071254221747926836, −1.21093995130486070183687447628, −1.19510657100474108319101604816, −1.06029860310480422841191834835, −0.929372061749576014449283424632, −0.68683491892878704477999872415, −0.25455259576145993904200879399,
0.25455259576145993904200879399, 0.68683491892878704477999872415, 0.929372061749576014449283424632, 1.06029860310480422841191834835, 1.19510657100474108319101604816, 1.21093995130486070183687447628, 1.82695044901071254221747926836, 2.08782257801348456766684296828, 2.09952992063649657576148441267, 2.13405825790575222701300227909, 2.16937215793739853827687520722, 2.20546741915855906334739078503, 2.51090266755196304743512274390, 2.81754000355144506521211399300, 3.25481103250109113844278881707, 3.33731053959229610234761154586, 3.34121664186695758072185403195, 3.68401546836073886454470142959, 3.95583534692726402967389595060, 4.00433357856061170614471890027, 4.00475554551239750582111179535, 4.35705263875738290081507153844, 4.36818163070755102673846689061, 4.50520459015240422442881368514, 4.92235331727106541157958683393
Plot not available for L-functions of degree greater than 10.
