| L(s) = 1 | + 2-s + 3·5-s + 7-s − 8-s + 3·10-s − 2·13-s + 14-s − 16-s + 3·17-s − 2·19-s + 5·25-s − 2·26-s − 8·31-s + 3·34-s + 3·35-s + 4·37-s − 2·38-s − 3·40-s + 3·41-s − 2·43-s + 3·47-s − 2·49-s + 5·50-s + 6·53-s − 56-s − 17·61-s − 8·62-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.34·5-s + 0.377·7-s − 0.353·8-s + 0.948·10-s − 0.554·13-s + 0.267·14-s − 1/4·16-s + 0.727·17-s − 0.458·19-s + 25-s − 0.392·26-s − 1.43·31-s + 0.514·34-s + 0.507·35-s + 0.657·37-s − 0.324·38-s − 0.474·40-s + 0.468·41-s − 0.304·43-s + 0.437·47-s − 2/7·49-s + 0.707·50-s + 0.824·53-s − 0.133·56-s − 2.17·61-s − 1.01·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.015918253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.015918253\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 60 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 - 3 T - 23 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 3 T - p T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 14 T + 156 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T - 39 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5 T + 81 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 124 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 13 T + 207 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
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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
−15.0924835796, −14.9008077648, −14.3842501853, −14.1183743309, −13.5380285186, −13.1762231336, −12.7508717346, −12.2206116198, −11.8245133300, −11.0798041997, −10.6297217493, −10.0885893224, −9.65254842649, −9.00502997176, −8.76427689750, −7.71763695356, −7.42743752596, −6.55195943646, −5.92179520590, −5.61837670395, −4.90070955562, −4.35428106074, −3.38224952032, −2.54241019486, −1.64738007120,
1.64738007120, 2.54241019486, 3.38224952032, 4.35428106074, 4.90070955562, 5.61837670395, 5.92179520590, 6.55195943646, 7.42743752596, 7.71763695356, 8.76427689750, 9.00502997176, 9.65254842649, 10.0885893224, 10.6297217493, 11.0798041997, 11.8245133300, 12.2206116198, 12.7508717346, 13.1762231336, 13.5380285186, 14.1183743309, 14.3842501853, 14.9008077648, 15.0924835796
