| L(s) = 1 | + 2-s + 4-s − 6·5-s + 8·7-s + 8-s − 5·9-s − 6·10-s + 8·11-s + 2·13-s + 8·14-s + 16-s + 2·17-s − 5·18-s − 2·19-s − 6·20-s + 8·22-s + 8·23-s + 18·25-s + 2·26-s + 8·28-s − 14·29-s − 4·31-s + 32-s + 2·34-s − 48·35-s − 5·36-s − 4·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 2.68·5-s + 3.02·7-s + 0.353·8-s − 5/3·9-s − 1.89·10-s + 2.41·11-s + 0.554·13-s + 2.13·14-s + 1/4·16-s + 0.485·17-s − 1.17·18-s − 0.458·19-s − 1.34·20-s + 1.70·22-s + 1.66·23-s + 18/5·25-s + 0.392·26-s + 1.51·28-s − 2.59·29-s − 0.718·31-s + 0.176·32-s + 0.342·34-s − 8.11·35-s − 5/6·36-s − 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.113759337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.113759337\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−14.3734863297, −13.9162987574, −13.4591802958, −12.3882574259, −12.2004291743, −11.6625684320, −11.6151508324, −11.1874716059, −10.9885026417, −10.7899417366, −9.14911318193, −8.95789599570, −8.40848690901, −8.22124144045, −7.63507836656, −7.25038064773, −6.81912640761, −5.77370446643, −5.28069074518, −4.81894959123, −4.02436568492, −3.86057015395, −3.41091529262, −2.01755248812, −1.10387421080,
1.10387421080, 2.01755248812, 3.41091529262, 3.86057015395, 4.02436568492, 4.81894959123, 5.28069074518, 5.77370446643, 6.81912640761, 7.25038064773, 7.63507836656, 8.22124144045, 8.40848690901, 8.95789599570, 9.14911318193, 10.7899417366, 10.9885026417, 11.1874716059, 11.6151508324, 11.6625684320, 12.2004291743, 12.3882574259, 13.4591802958, 13.9162987574, 14.3734863297
