| L(s) = 1 | − 3·4-s + 68·13-s + 25·16-s − 172·19-s − 114·25-s − 408·31-s + 1.32e3·37-s + 116·43-s − 204·52-s + 2.67e3·61-s − 315·64-s + 1.44e3·67-s + 396·73-s + 516·76-s + 1.22e3·79-s + 624·97-s + 342·100-s − 1.30e3·103-s + 720·109-s + 582·121-s + 1.22e3·124-s + 127-s + 131-s + 137-s + 139-s − 3.96e3·148-s + 149-s + ⋯ |
| L(s) = 1 | − 3/8·4-s + 1.45·13-s + 0.390·16-s − 2.07·19-s − 0.911·25-s − 2.36·31-s + 5.86·37-s + 0.411·43-s − 0.544·52-s + 5.60·61-s − 0.615·64-s + 2.63·67-s + 0.634·73-s + 0.778·76-s + 1.73·79-s + 0.653·97-s + 0.341·100-s − 1.24·103-s + 0.632·109-s + 0.437·121-s + 0.886·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 2.19·148-s + 0.000549·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.447076776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.447076776\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + 3 T^{2} - p^{4} T^{4} + 3 p^{6} T^{6} + p^{12} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 114 T^{2} + 25139 T^{4} + 114 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 582 T^{2} - 499957 T^{4} - 582 p^{6} T^{6} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 34 T - 582 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 9646 T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 86 T + 13227 T^{2} + 86 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 334 p T^{2} + 253878819 T^{4} + 334 p^{7} T^{6} + p^{12} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 79152 T^{2} + 2674078958 T^{4} + 79152 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 204 T + 55361 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 660 T + 204941 T^{2} - 660 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 4794 T^{2} + 6674554091 T^{4} + 4794 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 58 T + 159270 T^{2} - 58 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 19688 T^{2} - 8519622546 T^{4} + 19688 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 26852 T^{2} + 7143822294 T^{4} + 26852 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 603752 T^{2} + 171721941918 T^{4} + 603752 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 1336 T + 890826 T^{2} - 1336 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 722 T + 520662 T^{2} - 722 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 582818 T^{2} + 161061490563 T^{4} + 582818 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 198 T + 773210 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 610 T + 980238 T^{2} - 610 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 1579208 T^{2} + 1195208061054 T^{4} + 1579208 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 2521002 T^{2} + 2575576968923 T^{4} + 2521002 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 312 T + 1840322 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−6.56218254597625792967669205816, −6.36988191177389327727810923945, −5.91351902338233832174905264054, −5.77152848922439391695419823672, −5.67203426912053884853486149897, −5.58626466266196626954064722800, −5.18474163562930591268857888614, −4.83907158347323345339405052832, −4.74994257986462456538914680011, −4.30160803245176864651427985934, −4.07000984457363537420500899047, −4.03628000803736694765882855051, −3.81535765385454747994025872823, −3.57523787172336859884453511744, −3.47786240323766936653072639440, −2.79159577323772058150256617136, −2.58520971911799224825745820198, −2.47162418026906036769262843872, −2.06755728409460973295178542563, −1.97304883404062398980946370739, −1.58168507727422916655856884221, −0.941349331093177547140292894585, −0.917592372280660955496232910496, −0.49306999778494320234740779790, −0.46660114552583869593338181896,
0.46660114552583869593338181896, 0.49306999778494320234740779790, 0.917592372280660955496232910496, 0.941349331093177547140292894585, 1.58168507727422916655856884221, 1.97304883404062398980946370739, 2.06755728409460973295178542563, 2.47162418026906036769262843872, 2.58520971911799224825745820198, 2.79159577323772058150256617136, 3.47786240323766936653072639440, 3.57523787172336859884453511744, 3.81535765385454747994025872823, 4.03628000803736694765882855051, 4.07000984457363537420500899047, 4.30160803245176864651427985934, 4.74994257986462456538914680011, 4.83907158347323345339405052832, 5.18474163562930591268857888614, 5.58626466266196626954064722800, 5.67203426912053884853486149897, 5.77152848922439391695419823672, 5.91351902338233832174905264054, 6.36988191177389327727810923945, 6.56218254597625792967669205816
