| L(s) = 1 | − 2·2-s − 2·3-s − 5-s + 4·6-s − 3·7-s + 4·8-s + 2·9-s + 2·10-s − 3·11-s − 3·13-s + 6·14-s + 2·15-s − 4·16-s − 4·18-s − 3·19-s + 6·21-s + 6·22-s + 9·23-s − 8·24-s − 2·25-s + 6·26-s − 6·27-s + 5·29-s − 4·30-s − 11·31-s + 6·33-s + 3·35-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s − 0.447·5-s + 1.63·6-s − 1.13·7-s + 1.41·8-s + 2/3·9-s + 0.632·10-s − 0.904·11-s − 0.832·13-s + 1.60·14-s + 0.516·15-s − 16-s − 0.942·18-s − 0.688·19-s + 1.30·21-s + 1.27·22-s + 1.87·23-s − 1.63·24-s − 2/5·25-s + 1.17·26-s − 1.15·27-s + 0.928·29-s − 0.730·30-s − 1.97·31-s + 1.04·33-s + 0.507·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65767 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65767 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 5059 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 31 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 17 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 11 T + 67 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 11 T + 87 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 46 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T + 77 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 3 T + 11 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 3 T + 45 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 126 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + T - 109 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 116 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 66 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−15.0262459224, −14.6670981960, −13.9086484080, −13.2981725137, −13.0099955975, −12.7748167553, −12.1047766283, −11.7052438090, −11.1099362274, −10.6072052036, −10.3336985496, −9.88898937397, −9.36959526660, −9.04872790959, −8.42950315858, −7.98640902722, −7.33802010109, −6.90241906686, −6.48091344025, −5.55002213030, −5.08465936532, −4.70697502320, −3.69167276042, −3.04895983514, −1.68233473880, 0, 0,
1.68233473880, 3.04895983514, 3.69167276042, 4.70697502320, 5.08465936532, 5.55002213030, 6.48091344025, 6.90241906686, 7.33802010109, 7.98640902722, 8.42950315858, 9.04872790959, 9.36959526660, 9.88898937397, 10.3336985496, 10.6072052036, 11.1099362274, 11.7052438090, 12.1047766283, 12.7748167553, 13.0099955975, 13.2981725137, 13.9086484080, 14.6670981960, 15.0262459224
