| L(s) = 1 | + 2-s + 3-s + 2·5-s + 6-s − 2·7-s + 8-s + 9-s + 2·10-s − 4·13-s − 2·14-s + 2·15-s − 16-s + 6·17-s + 18-s + 2·19-s − 2·21-s − 4·23-s + 24-s + 2·25-s − 4·26-s + 4·27-s + 3·29-s + 2·30-s + 2·31-s − 6·32-s + 6·34-s − 4·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.894·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.10·13-s − 0.534·14-s + 0.516·15-s − 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.458·19-s − 0.436·21-s − 0.834·23-s + 0.204·24-s + 2/5·25-s − 0.784·26-s + 0.769·27-s + 0.557·29-s + 0.365·30-s + 0.359·31-s − 1.06·32-s + 1.02·34-s − 0.676·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22885 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22885 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.073819418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.073819418\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 3 T + p T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 3 T + p T^{2} ) \) |
| 199 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 20 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 17 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 11 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T + 64 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 64 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 - 15 T + 220 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T - 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−15.5979572558, −14.9965855443, −14.2555874866, −14.1196835923, −13.9760450468, −13.2827821622, −12.8168595611, −12.4194241345, −12.0330715696, −11.3886999465, −10.3778909816, −10.2163988644, −9.82081907368, −9.19158763296, −8.77296484832, −7.88310715068, −7.42805245213, −6.86669856107, −6.11531078514, −5.58398028417, −4.87314277325, −4.33080464239, −3.32526187217, −2.79697196714, −1.71834088674,
1.71834088674, 2.79697196714, 3.32526187217, 4.33080464239, 4.87314277325, 5.58398028417, 6.11531078514, 6.86669856107, 7.42805245213, 7.88310715068, 8.77296484832, 9.19158763296, 9.82081907368, 10.2163988644, 10.3778909816, 11.3886999465, 12.0330715696, 12.4194241345, 12.8168595611, 13.2827821622, 13.9760450468, 14.1196835923, 14.2555874866, 14.9965855443, 15.5979572558
