| L(s) = 1 | − 4·5-s − 4·7-s − 2·9-s + 8·11-s + 4·13-s + 8·19-s − 12·23-s + 10·25-s − 8·27-s − 4·29-s + 16·31-s + 16·35-s + 4·37-s − 12·41-s − 16·43-s + 8·45-s + 8·49-s + 4·53-s − 32·55-s + 16·59-s + 4·61-s + 8·63-s − 16·65-s + 8·67-s + 12·71-s + 28·73-s − 32·77-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 1.51·7-s − 2/3·9-s + 2.41·11-s + 1.10·13-s + 1.83·19-s − 2.50·23-s + 2·25-s − 1.53·27-s − 0.742·29-s + 2.87·31-s + 2.70·35-s + 0.657·37-s − 1.87·41-s − 2.43·43-s + 1.19·45-s + 8/7·49-s + 0.549·53-s − 4.31·55-s + 2.08·59-s + 0.512·61-s + 1.00·63-s − 1.98·65-s + 0.977·67-s + 1.42·71-s + 3.27·73-s − 3.64·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7096370182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7096370182\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + 2 T + p T^{2} )^{2}( 1 - 8 T^{2} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 8 T + 18 T^{2} + 160 T^{3} - 1246 T^{4} + 160 p T^{5} + 18 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 300 T^{3} + 1246 T^{4} + 300 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 4 T + 6 T^{2} - 204 T^{3} - 830 T^{4} - 204 p T^{5} + 6 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 516 T^{3} + 3694 T^{4} + 516 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 16 T + 162 T^{2} + 1384 T^{3} + 10178 T^{4} + 1384 p T^{5} + 162 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 486 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 4 T + 54 T^{2} - 708 T^{3} + 3490 T^{4} - 708 p T^{5} + 54 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 16 T + 114 T^{2} - 696 T^{3} + 4834 T^{4} - 696 p T^{5} + 114 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 8 T + 18 T^{2} + 736 T^{3} - 5854 T^{4} + 736 p T^{5} + 18 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 876 T^{3} + 10654 T^{4} - 876 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 16 T + 114 T^{2} + 792 T^{3} + 6370 T^{4} + 792 p T^{5} + 114 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 516 T^{3} + 1582 T^{4} - 516 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−9.834339661981159416906086060811, −9.743432227012172835864545995063, −9.345175935594503143631372857265, −9.046186220322066091333800194428, −8.440810548736777930964437681791, −8.384119826253477384487204040967, −8.251659939268628398375211144370, −8.102120985609435411616459689078, −7.48059168041753314794545315847, −7.26802842249833095936075354794, −6.66741511244973805435690844446, −6.65067413958643854251610485494, −6.58073189263355946620158030198, −6.02892857622969747385174629820, −5.80751545761279233116925922879, −5.30527144455619649660238974294, −4.98923570839777185369827673984, −4.32411170575179470425189897881, −3.92492067149424886385048659598, −3.78317858452596858180933677268, −3.50359414138225868731632229317, −3.36055108308096228739991492130, −2.62267687397708442152110336678, −1.78087375666559412677570346116, −0.802755396945065606216234866833,
0.802755396945065606216234866833, 1.78087375666559412677570346116, 2.62267687397708442152110336678, 3.36055108308096228739991492130, 3.50359414138225868731632229317, 3.78317858452596858180933677268, 3.92492067149424886385048659598, 4.32411170575179470425189897881, 4.98923570839777185369827673984, 5.30527144455619649660238974294, 5.80751545761279233116925922879, 6.02892857622969747385174629820, 6.58073189263355946620158030198, 6.65067413958643854251610485494, 6.66741511244973805435690844446, 7.26802842249833095936075354794, 7.48059168041753314794545315847, 8.102120985609435411616459689078, 8.251659939268628398375211144370, 8.384119826253477384487204040967, 8.440810548736777930964437681791, 9.046186220322066091333800194428, 9.345175935594503143631372857265, 9.743432227012172835864545995063, 9.834339661981159416906086060811
