| L(s) = 1 | + 2·5-s + 3·9-s − 10·17-s + 11·25-s − 32·29-s − 10·37-s + 16·41-s + 6·45-s − 2·49-s − 2·53-s + 22·61-s − 30·73-s + 9·81-s − 20·85-s − 14·89-s + 48·97-s − 22·101-s − 6·109-s + 16·113-s − 5·121-s + 38·125-s + 127-s + 131-s + 137-s + 139-s − 64·145-s + 149-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 9-s − 2.42·17-s + 11/5·25-s − 5.94·29-s − 1.64·37-s + 2.49·41-s + 0.894·45-s − 2/7·49-s − 0.274·53-s + 2.81·61-s − 3.51·73-s + 81-s − 2.16·85-s − 1.48·89-s + 4.87·97-s − 2.18·101-s − 0.574·109-s + 1.50·113-s − 0.454·121-s + 3.39·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.31·145-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.622242407\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.622242407\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \) |
| 5 | $C_2^2$ | \( ( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 5 T^{2} - 96 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 - 35 T^{2} + 864 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - 43 T^{2} + 1320 T^{4} - 43 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )( 1 + 59 T^{2} + p^{2} T^{4} ) \) |
| 37 | $C_2^2$ | \( ( 1 + 5 T - 12 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 19 T^{2} - 1848 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + T - 52 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 115 T^{2} + 9744 T^{4} - 115 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 109 T^{2} + p^{2} T^{4} )( 1 + 122 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 15 T + 152 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^3$ | \( 1 - 155 T^{2} + 17784 T^{4} - 155 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 + 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 7 T - 40 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
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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
−7.87157349854703519668036689047, −7.62944754832085948498509454762, −7.44619092933150430715551064251, −7.42905379687482957983230238156, −6.89775318776566433154205662380, −6.88255093177831673342426379696, −6.70854701317858839821165384533, −6.35483066505742571145785652903, −5.85868034271099397450013119908, −5.75083472431591313032915091470, −5.61128676605861189345793907248, −5.40044168577497510452821801160, −4.85983797560327033523500560990, −4.63224995481837020911237578452, −4.55303842784167273196598710644, −4.00968015610822571681707353223, −3.86031676145140149729509640890, −3.57133345901698184193825708254, −3.28332726545800663096597365968, −2.63240326008029503075519965577, −2.24539711565485849727439402070, −2.21075366775886574506629923410, −1.56004751391240996199052382506, −1.55436286452054426289403976863, −0.42684712419728731085954410879,
0.42684712419728731085954410879, 1.55436286452054426289403976863, 1.56004751391240996199052382506, 2.21075366775886574506629923410, 2.24539711565485849727439402070, 2.63240326008029503075519965577, 3.28332726545800663096597365968, 3.57133345901698184193825708254, 3.86031676145140149729509640890, 4.00968015610822571681707353223, 4.55303842784167273196598710644, 4.63224995481837020911237578452, 4.85983797560327033523500560990, 5.40044168577497510452821801160, 5.61128676605861189345793907248, 5.75083472431591313032915091470, 5.85868034271099397450013119908, 6.35483066505742571145785652903, 6.70854701317858839821165384533, 6.88255093177831673342426379696, 6.89775318776566433154205662380, 7.42905379687482957983230238156, 7.44619092933150430715551064251, 7.62944754832085948498509454762, 7.87157349854703519668036689047
