| 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 − 0.727·11-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(12.89236170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.89236170\) |
| \(L(2)\) |
|
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 | $D_4\times C_2$ | \( 1 - 320 T + 51200 T^{2} - 6683520 T^{3} + 740728463 T^{4} - 6683520 p^{2} T^{5} + 51200 p^{4} T^{6} - 320 p^{6} T^{7} + p^{8} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.69251491605744968392424955893, −7.15608564776996722058021381486, −6.83539263712905232830756300149, −6.77156138854792641975724714700, −6.74059370702176364455561503320, −5.99707032142572137861563293168, −5.95602548860890978841794088463, −5.91316601444185776173671017552, −5.70761931916456700990288971049, −5.19541297452887717667759474686, −4.92316549027702990729327825434, −4.79807141999028630286050191455, −4.67721602825181243574222086979, −3.90695098922698027938242351174, −3.88173759274242096777639134274, −3.84949674446251596134184917554, −3.48676679721458799471014036754, −2.87581195288427901548100638631, −2.75068257625205139214560050809, −2.35242428200595951349936936704, −2.22382334994836460291020928426, −1.33936140956052270359159840803, −0.996702155406713467577453029124, −0.924251665014973994632257173966, −0.892343155790627957989094404399,
0.892343155790627957989094404399, 0.924251665014973994632257173966, 0.996702155406713467577453029124, 1.33936140956052270359159840803, 2.22382334994836460291020928426, 2.35242428200595951349936936704, 2.75068257625205139214560050809, 2.87581195288427901548100638631, 3.48676679721458799471014036754, 3.84949674446251596134184917554, 3.88173759274242096777639134274, 3.90695098922698027938242351174, 4.67721602825181243574222086979, 4.79807141999028630286050191455, 4.92316549027702990729327825434, 5.19541297452887717667759474686, 5.70761931916456700990288971049, 5.91316601444185776173671017552, 5.95602548860890978841794088463, 5.99707032142572137861563293168, 6.74059370702176364455561503320, 6.77156138854792641975724714700, 6.83539263712905232830756300149, 7.15608564776996722058021381486, 7.69251491605744968392424955893
