| L(s) = 1 | + 6·7-s + 4·9-s − 4·17-s − 8·23-s − 3·25-s − 16·31-s − 36·41-s + 20·47-s + 21·49-s + 24·63-s − 48·71-s − 44·73-s + 16·79-s − 52·89-s + 28·97-s + 52·103-s − 4·113-s − 24·119-s + 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 16·153-s + 157-s + ⋯ |
| L(s) = 1 | + 2.26·7-s + 4/3·9-s − 0.970·17-s − 1.66·23-s − 3/5·25-s − 2.87·31-s − 5.62·41-s + 2.91·47-s + 3·49-s + 3.02·63-s − 5.69·71-s − 5.14·73-s + 1.80·79-s − 5.51·89-s + 2.84·97-s + 5.12·103-s − 0.376·113-s − 2.20·119-s + 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 1.29·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.789528171\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.789528171\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + T^{2} )^{3} \) |
| 7 | \( ( 1 - T )^{6} \) |
| good | 3 | \( 1 - 4 T^{2} + 16 T^{4} - 62 T^{6} + 16 p^{2} T^{8} - 4 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 28 T^{2} + 344 T^{4} - 3354 T^{6} + 344 p^{2} T^{8} - 28 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( 1 - 32 T^{2} + 64 p T^{4} - 11846 T^{6} + 64 p^{3} T^{8} - 32 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 + 2 T - 4 T^{2} - 38 T^{3} - 4 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( 1 - 58 T^{2} + 1943 T^{4} - 44652 T^{6} + 1943 p^{2} T^{8} - 58 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 + 4 T + 9 T^{2} + 24 T^{3} + 9 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( 1 - 72 T^{2} + 4136 T^{4} - 132218 T^{6} + 4136 p^{2} T^{8} - 72 p^{4} T^{10} + p^{6} T^{12} \) |
| 31 | \( ( 1 + 8 T + 69 T^{2} + 464 T^{3} + 69 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 54 T^{2} + 167 T^{4} + 59788 T^{6} + 167 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 18 T + 203 T^{2} + 1556 T^{3} + 203 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{3} \) |
| 47 | \( ( 1 - 10 T + 154 T^{2} - 876 T^{3} + 154 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( 1 - 142 T^{2} + 11543 T^{4} - 697956 T^{6} + 11543 p^{2} T^{8} - 142 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( 1 - 178 T^{2} + 17399 T^{4} - 1199004 T^{6} + 17399 p^{2} T^{8} - 178 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 - 74 T^{2} + 823 T^{4} + 21940 T^{6} + 823 p^{2} T^{8} - 74 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( 1 - 238 T^{2} + 29207 T^{4} - 2378244 T^{6} + 29207 p^{2} T^{8} - 238 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 24 T + 377 T^{2} + 3728 T^{3} + 377 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 + 22 T + 315 T^{2} + 3052 T^{3} + 315 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( ( 1 - 8 T + 58 T^{2} + 208 T^{3} + 58 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 442 T^{2} + 85527 T^{4} - 9250156 T^{6} + 85527 p^{2} T^{8} - 442 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 26 T + 427 T^{2} + 4788 T^{3} + 427 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( ( 1 - 14 T + 300 T^{2} - 2486 T^{3} + 300 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
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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
−4.56274799062625779990363990089, −4.51063219176026221838959536663, −4.44296822695948585685345184130, −4.39358595914105853107362207497, −4.16655448808813890231176758770, −3.94120775344693292309701218904, −3.83588781176178688728788467731, −3.77979129091918002243637389255, −3.50871232950083431880120311029, −3.32987285354864664876008043185, −3.12031619969504847086037875348, −3.03402994222485256112316961789, −2.87186210680163985930002467226, −2.52946718561346912313985663563, −2.33026960688443924944036738347, −2.28629000460414035680861320333, −1.84371465076296970886825925587, −1.72843529028177038247500434392, −1.63609235925581354553309748732, −1.58592346521942122119711418722, −1.57043453676152056929602360160, −1.40673773925420013214154843783, −0.67496828827048093893291980602, −0.52180139971445660673906628153, −0.15570359895846342185710648364,
0.15570359895846342185710648364, 0.52180139971445660673906628153, 0.67496828827048093893291980602, 1.40673773925420013214154843783, 1.57043453676152056929602360160, 1.58592346521942122119711418722, 1.63609235925581354553309748732, 1.72843529028177038247500434392, 1.84371465076296970886825925587, 2.28629000460414035680861320333, 2.33026960688443924944036738347, 2.52946718561346912313985663563, 2.87186210680163985930002467226, 3.03402994222485256112316961789, 3.12031619969504847086037875348, 3.32987285354864664876008043185, 3.50871232950083431880120311029, 3.77979129091918002243637389255, 3.83588781176178688728788467731, 3.94120775344693292309701218904, 4.16655448808813890231176758770, 4.39358595914105853107362207497, 4.44296822695948585685345184130, 4.51063219176026221838959536663, 4.56274799062625779990363990089
Plot not available for L-functions of degree greater than 10.
