| L(s) = 1 | − 8·7-s + 8·11-s + 48·13-s + 8·17-s − 72·23-s − 44·31-s + 112·37-s − 32·41-s − 104·43-s − 80·47-s + 32·49-s + 104·53-s − 180·61-s − 264·67-s − 256·71-s + 112·73-s − 64·77-s − 9·81-s + 16·83-s − 384·91-s + 320·97-s − 496·101-s − 144·103-s + 264·107-s + 32·113-s − 64·119-s − 396·121-s + ⋯ |
| L(s) = 1 | − 8/7·7-s + 8/11·11-s + 3.69·13-s + 8/17·17-s − 3.13·23-s − 1.41·31-s + 3.02·37-s − 0.780·41-s − 2.41·43-s − 1.70·47-s + 0.653·49-s + 1.96·53-s − 2.95·61-s − 3.94·67-s − 3.60·71-s + 1.53·73-s − 0.831·77-s − 1/9·81-s + 0.192·83-s − 4.21·91-s + 3.29·97-s − 4.91·101-s − 1.39·103-s + 2.46·107-s + 0.283·113-s − 0.537·119-s − 3.27·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4172072505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4172072505\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 432 T^{3} + 5807 T^{4} + 432 p^{2} T^{5} + 32 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 222 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 48 T + 1152 T^{2} - 18336 T^{3} + 246479 T^{4} - 18336 p^{2} T^{5} + 1152 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 2280 T^{3} + 162434 T^{4} - 2280 p^{2} T^{5} + 32 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 1346 T^{2} + 711171 T^{4} - 1346 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 72 T + 2592 T^{2} + 76968 T^{3} + 1993922 T^{4} + 76968 p^{2} T^{5} + 2592 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1004 T^{2} + 1647750 T^{4} - 1004 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 22 T + 1827 T^{2} + 22 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 112 T + 6272 T^{2} - 280560 T^{3} + 11259554 T^{4} - 280560 p^{2} T^{5} + 6272 p^{4} T^{6} - 112 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 16 T + 3402 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 104 T + 5408 T^{2} + 220272 T^{3} + 8899487 T^{4} + 220272 p^{2} T^{5} + 5408 p^{4} T^{6} + 104 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 80 T + 3200 T^{2} + 193680 T^{3} + 11677538 T^{4} + 193680 p^{2} T^{5} + 3200 p^{4} T^{6} + 80 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 104 T + 5408 T^{2} - 113256 T^{3} - 586558 T^{4} - 113256 p^{2} T^{5} + 5408 p^{4} T^{6} - 104 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 1628 T^{2} + 24799014 T^{4} - 1628 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 90 T + 6011 T^{2} + 90 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 264 T + 34848 T^{2} + 3306864 T^{3} + 249207983 T^{4} + 3306864 p^{2} T^{5} + 34848 p^{4} T^{6} + 264 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 128 T + 13002 T^{2} + 128 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 112 T + 6272 T^{2} - 638064 T^{3} + 64776194 T^{4} - 638064 p^{2} T^{5} + 6272 p^{4} T^{6} - 112 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 12604 T^{2} + 115279110 T^{4} - 12604 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} + 124464 T^{3} - 94124542 T^{4} + 124464 p^{2} T^{5} + 128 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 20708 T^{2} + 217938054 T^{4} - 20708 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 320 T + 51200 T^{2} - 6683520 T^{3} + 740728463 T^{4} - 6683520 p^{2} T^{5} + 51200 p^{4} T^{6} - 320 p^{6} T^{7} + p^{8} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.59688514291544328488863509801, −6.37174733711220848196595566408, −6.30914226145723935735824519479, −6.11485160325057765988790158427, −5.90321134072117137282059336127, −5.83701680279340757981516828622, −5.81387707754950799651409222948, −5.10723166357050988289221628564, −4.95005816462521775049791746631, −4.82130953414174271897361389793, −4.28077250410374239958765724571, −3.94690575462706429459807059093, −3.89610533643209519376938058670, −3.87001633526596544159031368273, −3.66558367665814628433992545070, −3.27600747922711905009195735092, −2.92832272943218759079811889149, −2.70379324117270388106961852036, −2.60914190729332968386700622030, −1.72175750309240181507364025904, −1.54337969002848016527097969691, −1.44491148853257613938960752596, −1.40969149857387187415047990993, −0.57429282979663121938557124706, −0.10396537190795019219888463188,
0.10396537190795019219888463188, 0.57429282979663121938557124706, 1.40969149857387187415047990993, 1.44491148853257613938960752596, 1.54337969002848016527097969691, 1.72175750309240181507364025904, 2.60914190729332968386700622030, 2.70379324117270388106961852036, 2.92832272943218759079811889149, 3.27600747922711905009195735092, 3.66558367665814628433992545070, 3.87001633526596544159031368273, 3.89610533643209519376938058670, 3.94690575462706429459807059093, 4.28077250410374239958765724571, 4.82130953414174271897361389793, 4.95005816462521775049791746631, 5.10723166357050988289221628564, 5.81387707754950799651409222948, 5.83701680279340757981516828622, 5.90321134072117137282059336127, 6.11485160325057765988790158427, 6.30914226145723935735824519479, 6.37174733711220848196595566408, 6.59688514291544328488863509801
