| L(s) = 1 | − 2-s − 3-s − 2·5-s + 6-s − 3·7-s − 8-s + 9-s + 2·10-s − 3·11-s − 3·13-s + 3·14-s + 2·15-s − 16-s − 2·17-s − 18-s − 3·19-s + 3·21-s + 3·22-s + 24-s − 25-s + 3·26-s − 27-s + 29-s − 2·30-s − 8·31-s + 6·32-s + 3·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 0.894·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.904·11-s − 0.832·13-s + 0.801·14-s + 0.516·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.688·19-s + 0.654·21-s + 0.639·22-s + 0.204·24-s − 1/5·25-s + 0.588·26-s − 0.192·27-s + 0.185·29-s − 0.365·30-s − 1.43·31-s + 1.06·32-s + 0.522·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115047 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115047 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( 1 + T \) |
| 4261 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 45 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - T + 8 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 85 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 67 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - T + 79 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 10 T + 56 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T - 50 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T - 32 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3 T + 25 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T - 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + 2 T - 81 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 62 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−14.5146904495, −13.7943090537, −13.2379463301, −13.0997595791, −12.4349863654, −12.2579828825, −11.7391556936, −11.1664250281, −10.9534491976, −10.2043538639, −9.97328532283, −9.55753747859, −8.92036856907, −8.62955202586, −7.96650246138, −7.59012148254, −6.93811520007, −6.60675826010, −6.10878949600, −5.27350818727, −4.90578244307, −4.14942745481, −3.49701290206, −2.84826082341, −1.95460711371, 0, 0,
1.95460711371, 2.84826082341, 3.49701290206, 4.14942745481, 4.90578244307, 5.27350818727, 6.10878949600, 6.60675826010, 6.93811520007, 7.59012148254, 7.96650246138, 8.62955202586, 8.92036856907, 9.55753747859, 9.97328532283, 10.2043538639, 10.9534491976, 11.1664250281, 11.7391556936, 12.2579828825, 12.4349863654, 13.0997595791, 13.2379463301, 13.7943090537, 14.5146904495
