| L(s) = 1 | + 8·11-s + 8·13-s + 16·23-s + 12·25-s − 8·37-s − 16·47-s + 28·49-s + 16·59-s − 24·61-s + 16·71-s + 40·83-s − 24·97-s + 48·107-s + 40·109-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 64·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 20·169-s + 173-s + ⋯ |
| L(s) = 1 | + 2.41·11-s + 2.21·13-s + 3.33·23-s + 12/5·25-s − 1.31·37-s − 2.33·47-s + 4·49-s + 2.08·59-s − 3.07·61-s + 1.89·71-s + 4.39·83-s − 2.43·97-s + 4.64·107-s + 3.83·109-s + 3.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.35·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.53·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(136.2965669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(136.2965669\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( ( 1 - T )^{8} \) |
| good | 5 | \( 1 - 12 T^{2} + 16 p T^{4} - 372 T^{6} + 1614 T^{8} - 372 p^{2} T^{10} + 16 p^{5} T^{12} - 12 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( 1 - 4 p T^{2} + 416 T^{4} - 4260 T^{6} + 33422 T^{8} - 4260 p^{2} T^{10} + 416 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \) |
| 13 | \( ( 1 - 4 T + 14 T^{2} + 36 T^{3} - 154 T^{4} + 36 p T^{5} + 14 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 48 T^{2} + 1724 T^{4} - 42960 T^{6} + 826374 T^{8} - 42960 p^{2} T^{10} + 1724 p^{4} T^{12} - 48 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( 1 - 40 T^{2} + 1652 T^{4} - 39864 T^{6} + 928262 T^{8} - 39864 p^{2} T^{10} + 1652 p^{4} T^{12} - 40 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 8 T + 66 T^{2} - 336 T^{3} + 2086 T^{4} - 336 p T^{5} + 66 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( 1 - 160 T^{2} + 11420 T^{4} - 506208 T^{6} + 17124038 T^{8} - 506208 p^{2} T^{10} + 11420 p^{4} T^{12} - 160 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( ( 1 + 4 T + 56 T^{2} + 204 T^{3} + 2606 T^{4} + 204 p T^{5} + 56 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 - 240 T^{2} + 26300 T^{4} - 1782288 T^{6} + 85159878 T^{8} - 1782288 p^{2} T^{10} + 26300 p^{4} T^{12} - 240 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( 1 - 104 T^{2} + 7924 T^{4} - 453368 T^{6} + 22010950 T^{8} - 453368 p^{2} T^{10} + 7924 p^{4} T^{12} - 104 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( ( 1 + 8 T + 42 T^{2} + 576 T^{3} + 5398 T^{4} + 576 p T^{5} + 42 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 - 204 T^{2} + 23504 T^{4} - 1838388 T^{6} + 110812110 T^{8} - 1838388 p^{2} T^{10} + 23504 p^{4} T^{12} - 204 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 8 T + 108 T^{2} - 72 T^{3} + 2902 T^{4} - 72 p T^{5} + 108 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 12 T + 254 T^{2} + 2148 T^{3} + 23526 T^{4} + 2148 p T^{5} + 254 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 100 T^{2} + 8022 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 8 T + 258 T^{2} - 1488 T^{3} + 26662 T^{4} - 1488 p T^{5} + 258 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 140 T^{2} + 768 T^{3} + 9030 T^{4} + 768 p T^{5} + 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( 1 - 284 T^{2} + 48928 T^{4} - 5986532 T^{6} + 538243534 T^{8} - 5986532 p^{2} T^{10} + 48928 p^{4} T^{12} - 284 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( ( 1 - 20 T + 408 T^{2} - 4548 T^{3} + 51790 T^{4} - 4548 p T^{5} + 408 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 - 384 T^{2} + 62276 T^{4} - 5930112 T^{6} + 485181510 T^{8} - 5930112 p^{2} T^{10} + 62276 p^{4} T^{12} - 384 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 + 12 T + 368 T^{2} + 3348 T^{3} + 52590 T^{4} + 3348 p T^{5} + 368 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.33889789675306801419316239265, −3.19511122653511172443350006756, −2.96460913214935629444419482476, −2.88661795612716088050685268197, −2.87957866905549684559885470453, −2.85823345592559654608029805881, −2.63362607431833507333600951037, −2.57781006846515289783907127659, −2.55388101684956775118197880807, −2.15983456637838546572501849317, −1.96119921551081672734610617960, −1.90989514499797520576873265290, −1.77698309385182677774650405321, −1.73534572111326174726991382080, −1.68471671502702149391761144999, −1.65275731034834015410615197178, −1.51027461967043496225278876161, −1.09402804966479524087353108164, −0.971227525078215168247536401046, −0.958396921994699882290041829666, −0.825812954928076789593491977774, −0.73655421591423925575656701931, −0.51891173725560737107263238274, −0.51468862985453817488257688613, −0.50385806728382670605820336777,
0.50385806728382670605820336777, 0.51468862985453817488257688613, 0.51891173725560737107263238274, 0.73655421591423925575656701931, 0.825812954928076789593491977774, 0.958396921994699882290041829666, 0.971227525078215168247536401046, 1.09402804966479524087353108164, 1.51027461967043496225278876161, 1.65275731034834015410615197178, 1.68471671502702149391761144999, 1.73534572111326174726991382080, 1.77698309385182677774650405321, 1.90989514499797520576873265290, 1.96119921551081672734610617960, 2.15983456637838546572501849317, 2.55388101684956775118197880807, 2.57781006846515289783907127659, 2.63362607431833507333600951037, 2.85823345592559654608029805881, 2.87957866905549684559885470453, 2.88661795612716088050685268197, 2.96460913214935629444419482476, 3.19511122653511172443350006756, 3.33889789675306801419316239265
Plot not available for L-functions of degree greater than 10.
