| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s + 2·5-s − 4·6-s − 3·7-s + 4·8-s + 3·9-s + 4·10-s − 2·11-s − 6·12-s − 13-s − 6·14-s − 4·15-s + 5·16-s + 2·17-s + 6·18-s + 5·19-s + 6·20-s + 6·21-s − 4·22-s − 3·23-s − 8·24-s + 3·25-s − 2·26-s − 4·27-s − 9·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s − 1.13·7-s + 1.41·8-s + 9-s + 1.26·10-s − 0.603·11-s − 1.73·12-s − 0.277·13-s − 1.60·14-s − 1.03·15-s + 5/4·16-s + 0.485·17-s + 1.41·18-s + 1.14·19-s + 1.34·20-s + 1.30·21-s − 0.852·22-s − 0.625·23-s − 1.63·24-s + 3/5·25-s − 0.392·26-s − 0.769·27-s − 1.70·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.168509056\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.168509056\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T + 16 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 34 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 54 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T + 64 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 62 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 11 T + 154 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 10 T + 126 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 130 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 126 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 11 T + 186 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 11 T + 132 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.902869649978967835915518669564, −7.75975980227104019273101264985, −7.44130626660409044515847737195, −7.17983409218493472169887882074, −6.59790417246279258986775035236, −6.44346289893681421474503359851, −5.87504534385872603107896358779, −5.79695363857826765696681975757, −5.62569875596264527145131507213, −5.13428426084232158922348005088, −4.82308122537143305295434618145, −4.37671444970624292750185887487, −3.77889132110045473631274127554, −3.73964035749030762150726116033, −2.96195183225469407336059706662, −2.80804869321583419550457842638, −2.05617421545381198950561832704, −1.97615875077853964639146140850, −0.890868132556719591976220717401, −0.67480439547222293119396811379,
0.67480439547222293119396811379, 0.890868132556719591976220717401, 1.97615875077853964639146140850, 2.05617421545381198950561832704, 2.80804869321583419550457842638, 2.96195183225469407336059706662, 3.73964035749030762150726116033, 3.77889132110045473631274127554, 4.37671444970624292750185887487, 4.82308122537143305295434618145, 5.13428426084232158922348005088, 5.62569875596264527145131507213, 5.79695363857826765696681975757, 5.87504534385872603107896358779, 6.44346289893681421474503359851, 6.59790417246279258986775035236, 7.17983409218493472169887882074, 7.44130626660409044515847737195, 7.75975980227104019273101264985, 7.902869649978967835915518669564
