| L(s) = 1 | − 16·3-s + 96·9-s + 48·11-s + 80·17-s − 560·19-s − 156·25-s − 176·27-s − 768·33-s + 952·41-s − 432·43-s + 52·49-s − 1.28e3·51-s + 8.96e3·57-s + 16·59-s + 272·67-s + 1.76e3·73-s + 2.49e3·75-s − 562·81-s + 2.48e3·83-s + 1.44e3·89-s + 2.41e3·97-s + 4.60e3·99-s − 2.89e3·107-s + 1.88e3·113-s − 320·121-s − 1.52e4·123-s + 127-s + ⋯ |
| L(s) = 1 | − 3.07·3-s + 32/9·9-s + 1.31·11-s + 1.14·17-s − 6.76·19-s − 1.24·25-s − 1.25·27-s − 4.05·33-s + 3.62·41-s − 1.53·43-s + 0.151·49-s − 3.51·51-s + 20.8·57-s + 0.0353·59-s + 0.495·67-s + 2.82·73-s + 3.84·75-s − 0.770·81-s + 3.27·83-s + 1.71·89-s + 2.52·97-s + 4.67·99-s − 2.61·107-s + 1.56·113-s − 0.240·121-s − 11.1·123-s + 0.000698·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6066094830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6066094830\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( ( 1 + 8 T + 16 p T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 + 156 T^{2} + 14806 T^{4} + 156 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 213446 T^{4} - 52 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 24 T + 1024 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 4220 T^{2} + 5918 p^{3} T^{4} + 4220 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 40 T + 8818 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 280 T + 33296 T^{2} + 280 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 20620 T^{2} + 207523206 T^{4} + 20620 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 11580 T^{2} - 480509258 T^{4} + 11580 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 43836 T^{2} + 1862878214 T^{4} + 43836 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 125212 T^{2} + 7910484054 T^{4} + 125212 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 238 T + p^{3} T^{2} )^{4} \) |
| 43 | $D_{4}$ | \( ( 1 + 216 T + 170480 T^{2} + 216 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 322876 T^{2} + 47413040454 T^{4} + 322876 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 261660 T^{2} + 59349543958 T^{4} + 261660 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 8 T + 293536 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 898492 T^{2} + 304846294326 T^{4} + 898492 p^{6} T^{6} + p^{12} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 136 T - 52288 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 17108 p T^{2} + 622466215110 T^{4} + 17108 p^{7} T^{6} + p^{12} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 880 T + 654834 T^{2} - 880 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 986492 T^{2} + 697242115910 T^{4} + 986492 p^{6} T^{6} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 1240 T + 1524256 T^{2} - 1240 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 720 T + 1369170 T^{2} - 720 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 1208 T + 1445330 T^{2} - 1208 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−7.29968215020007534626160324155, −6.85204537572159108200975128719, −6.67348043557156076080513248524, −6.52944205409632806011175540899, −6.44738784661545298789924740477, −6.06200870652719411885991756647, −5.98070399252606223990117018328, −5.85501069043159840155619940825, −5.79829600383047609134611165599, −5.16266309275195599363542470368, −4.95293281647207126740079571017, −4.74150656790371910999434553566, −4.45761604988537812278633731788, −4.27184468437409879606117377566, −3.83960454197428705395112995134, −3.67590954518820733104335670349, −3.63226971483827101216694406734, −2.71439493798218086274732677457, −2.26756724690302123273618989500, −2.07418535104778156235308602193, −2.02403485884367126738557934109, −1.33631274299515937273930190580, −0.65681245450044742944107879089, −0.48085392411344382587300748032, −0.33003132777592774114151941500,
0.33003132777592774114151941500, 0.48085392411344382587300748032, 0.65681245450044742944107879089, 1.33631274299515937273930190580, 2.02403485884367126738557934109, 2.07418535104778156235308602193, 2.26756724690302123273618989500, 2.71439493798218086274732677457, 3.63226971483827101216694406734, 3.67590954518820733104335670349, 3.83960454197428705395112995134, 4.27184468437409879606117377566, 4.45761604988537812278633731788, 4.74150656790371910999434553566, 4.95293281647207126740079571017, 5.16266309275195599363542470368, 5.79829600383047609134611165599, 5.85501069043159840155619940825, 5.98070399252606223990117018328, 6.06200870652719411885991756647, 6.44738784661545298789924740477, 6.52944205409632806011175540899, 6.67348043557156076080513248524, 6.85204537572159108200975128719, 7.29968215020007534626160324155
