| L(s) = 1 | + 2-s − 3-s + 4·5-s − 6-s − 2·7-s + 4·10-s + 4·13-s − 2·14-s − 4·15-s − 2·17-s + 2·21-s − 24·23-s + 5·25-s + 4·26-s + 10·29-s − 4·30-s + 8·31-s − 32-s − 2·34-s − 8·35-s + 2·37-s − 4·39-s + 2·41-s + 2·42-s − 16·43-s − 24·46-s + 2·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1.78·5-s − 0.408·6-s − 0.755·7-s + 1.26·10-s + 1.10·13-s − 0.534·14-s − 1.03·15-s − 0.485·17-s + 0.436·21-s − 5.00·23-s + 25-s + 0.784·26-s + 1.85·29-s − 0.730·30-s + 1.43·31-s − 0.176·32-s − 0.342·34-s − 1.35·35-s + 0.328·37-s − 0.640·39-s + 0.312·41-s + 0.308·42-s − 2.43·43-s − 3.53·46-s + 0.291·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5168121105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5168121105\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 3 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | | \( 1 \) |
| good | 5 | $C_4\times C_2$ | \( 1 - 4 T + 11 T^{2} - 24 T^{3} + 41 T^{4} - 24 p T^{5} + 11 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 + 2 T - 3 T^{2} - 20 T^{3} - 19 T^{4} - 20 p T^{5} - 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_4\times C_2$ | \( 1 - 4 T + 3 T^{2} + 40 T^{3} - 199 T^{4} + 40 p T^{5} + 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 + 2 T - 13 T^{2} - 60 T^{3} + 101 T^{4} - 60 p T^{5} - 13 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 29 | $C_4\times C_2$ | \( 1 - 10 T + 71 T^{2} - 420 T^{3} + 2141 T^{4} - 420 p T^{5} + 71 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 - 8 T + 33 T^{2} - 16 T^{3} - 895 T^{4} - 16 p T^{5} + 33 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 - 2 T - 33 T^{2} + 140 T^{3} + 941 T^{4} + 140 p T^{5} - 33 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_4\times C_2$ | \( 1 - 2 T - 37 T^{2} + 156 T^{3} + 1205 T^{4} + 156 p T^{5} - 37 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 47 | $C_4\times C_2$ | \( 1 - 2 T - 43 T^{2} + 180 T^{3} + 1661 T^{4} + 180 p T^{5} - 43 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_4\times C_2$ | \( 1 + 4 T - 37 T^{2} - 360 T^{3} + 521 T^{4} - 360 p T^{5} - 37 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 61 | $C_4\times C_2$ | \( 1 + 8 T + 3 T^{2} - 464 T^{3} - 3895 T^{4} - 464 p T^{5} + 3 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 71 | $C_4\times C_2$ | \( 1 + 2 T - 67 T^{2} - 276 T^{3} + 4205 T^{4} - 276 p T^{5} - 67 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_4\times C_2$ | \( 1 + 6 T - 37 T^{2} - 660 T^{3} - 1259 T^{4} - 660 p T^{5} - 37 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 - 10 T + 21 T^{2} + 580 T^{3} - 7459 T^{4} + 580 p T^{5} + 21 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_4\times C_2$ | \( 1 - 4 T - 67 T^{2} + 600 T^{3} + 3161 T^{4} + 600 p T^{5} - 67 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 97 | $C_4\times C_2$ | \( 1 - 2 T - 93 T^{2} + 380 T^{3} + 8261 T^{4} + 380 p T^{5} - 93 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.56162304763252386938252625452, −7.25277901514010945405430483662, −6.56437182306160841128102421769, −6.48136389154395999889901316683, −6.38038832909039975290296936574, −6.37017077622831097669911532029, −6.09055987020567925177616939595, −5.92757519672131863867731563520, −5.78834186648881317117218020333, −5.27835766394675771721676027711, −5.16843178189019980253817251524, −4.94616323189077194123377416953, −4.55009859342003340383578753982, −4.22592083276388511284518728061, −4.13503757069248099976826635661, −3.77584003588981457796164081637, −3.72243869951311826228427089152, −3.07079915875883433441105871574, −2.85296352904313173204277465058, −2.43191550899103218833569121049, −2.43046996038888647250384531366, −1.67551128374113046277315321342, −1.58710591262511006362101752307, −1.31343638831302220106744895570, −0.16136384653226340996571649667,
0.16136384653226340996571649667, 1.31343638831302220106744895570, 1.58710591262511006362101752307, 1.67551128374113046277315321342, 2.43046996038888647250384531366, 2.43191550899103218833569121049, 2.85296352904313173204277465058, 3.07079915875883433441105871574, 3.72243869951311826228427089152, 3.77584003588981457796164081637, 4.13503757069248099976826635661, 4.22592083276388511284518728061, 4.55009859342003340383578753982, 4.94616323189077194123377416953, 5.16843178189019980253817251524, 5.27835766394675771721676027711, 5.78834186648881317117218020333, 5.92757519672131863867731563520, 6.09055987020567925177616939595, 6.37017077622831097669911532029, 6.38038832909039975290296936574, 6.48136389154395999889901316683, 6.56437182306160841128102421769, 7.25277901514010945405430483662, 7.56162304763252386938252625452
