| L(s) = 1 | − 4·3-s − 4-s + 2·7-s + 10·9-s + 4·12-s + 16-s − 2·19-s − 8·21-s − 25-s − 20·27-s − 2·28-s − 10·36-s − 2·43-s − 4·48-s + 3·49-s + 8·57-s + 20·63-s − 2·64-s − 2·67-s + 4·75-s + 2·76-s + 8·79-s + 35·81-s + 8·84-s + 2·97-s + 100-s − 4·103-s + ⋯ |
| L(s) = 1 | − 4·3-s − 4-s + 2·7-s + 10·9-s + 4·12-s + 16-s − 2·19-s − 8·21-s − 25-s − 20·27-s − 2·28-s − 10·36-s − 2·43-s − 4·48-s + 3·49-s + 8·57-s + 20·63-s − 2·64-s − 2·67-s + 4·75-s + 2·76-s + 8·79-s + 35·81-s + 8·84-s + 2·97-s + 100-s − 4·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 103^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 103^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1036942162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1036942162\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 103 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_1$ | \( ( 1 - T )^{8} \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
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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
−8.859475784895736116147925507584, −8.272709699092472690754421307092, −8.013656112543325759817886102575, −8.005020077092955937159925138564, −7.81738152233174997927230057950, −7.32919879401694972438922801721, −7.14524733030878046177574082744, −6.81183079253806388248110995934, −6.50161125793633505369815806766, −6.37633584613795819102729883788, −5.95833492645641223916834028934, −5.91955602453838400500700878125, −5.57716064173532486415388486645, −5.11057908106319239006716614245, −5.06384152626733809355963094977, −4.91233275782243722896651321198, −4.56737925639594246279429331414, −4.39624143448930314733982286000, −4.10005663227739509265934877482, −3.68952409233427246372690665657, −3.54216772932969819530359716818, −2.12258877966992706501097139339, −1.99031898283215354411522817944, −1.53386756588086862297515788310, −0.907851898509958435974744969953,
0.907851898509958435974744969953, 1.53386756588086862297515788310, 1.99031898283215354411522817944, 2.12258877966992706501097139339, 3.54216772932969819530359716818, 3.68952409233427246372690665657, 4.10005663227739509265934877482, 4.39624143448930314733982286000, 4.56737925639594246279429331414, 4.91233275782243722896651321198, 5.06384152626733809355963094977, 5.11057908106319239006716614245, 5.57716064173532486415388486645, 5.91955602453838400500700878125, 5.95833492645641223916834028934, 6.37633584613795819102729883788, 6.50161125793633505369815806766, 6.81183079253806388248110995934, 7.14524733030878046177574082744, 7.32919879401694972438922801721, 7.81738152233174997927230057950, 8.005020077092955937159925138564, 8.013656112543325759817886102575, 8.272709699092472690754421307092, 8.859475784895736116147925507584
