| L(s) = 1 | + 2-s + 2·3-s + 2·5-s + 2·6-s − 4·7-s + 8-s + 3·9-s + 2·10-s + 11-s − 3·13-s − 4·14-s + 4·15-s − 16-s + 2·17-s + 3·18-s − 6·19-s − 8·21-s + 22-s + 23-s + 2·24-s − 3·25-s − 3·26-s + 4·27-s + 6·29-s + 4·30-s − 6·32-s + 2·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 0.894·5-s + 0.816·6-s − 1.51·7-s + 0.353·8-s + 9-s + 0.632·10-s + 0.301·11-s − 0.832·13-s − 1.06·14-s + 1.03·15-s − 1/4·16-s + 0.485·17-s + 0.707·18-s − 1.37·19-s − 1.74·21-s + 0.213·22-s + 0.208·23-s + 0.408·24-s − 3/5·25-s − 0.588·26-s + 0.769·27-s + 1.11·29-s + 0.730·30-s − 1.06·32-s + 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18369 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18369 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.167012371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.167012371\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 157 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 17 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T - 14 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 7 T + 72 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 74 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 9 T + 106 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T - 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 16 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 53 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 104 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 10 T + 52 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−15.8752532012, −15.0906442612, −14.7552013523, −14.4254764001, −13.8036277540, −13.4923989184, −13.1244944604, −12.6711122000, −12.4012310830, −11.5609387940, −10.8185391160, −10.1128960882, −9.81212108590, −9.47703676684, −8.88556503523, −8.26089853509, −7.56306915375, −6.90103518451, −6.36023783079, −5.88103899427, −4.81516798350, −4.31541108796, −3.49180624754, −2.76268731882, −1.99785160301,
1.99785160301, 2.76268731882, 3.49180624754, 4.31541108796, 4.81516798350, 5.88103899427, 6.36023783079, 6.90103518451, 7.56306915375, 8.26089853509, 8.88556503523, 9.47703676684, 9.81212108590, 10.1128960882, 10.8185391160, 11.5609387940, 12.4012310830, 12.6711122000, 13.1244944604, 13.4923989184, 13.8036277540, 14.4254764001, 14.7552013523, 15.0906442612, 15.8752532012
