| L(s) = 1 | + 2-s + 3-s + 4-s − 3·5-s + 6-s + 8-s − 2·9-s − 3·10-s + 12-s − 3·15-s + 16-s − 2·18-s − 2·19-s − 3·20-s + 6·23-s + 24-s − 25-s − 5·27-s + 12·29-s − 3·30-s + 32-s − 2·36-s − 2·38-s − 3·40-s + 7·43-s + 6·45-s + 6·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.353·8-s − 2/3·9-s − 0.948·10-s + 0.288·12-s − 0.774·15-s + 1/4·16-s − 0.471·18-s − 0.458·19-s − 0.670·20-s + 1.25·23-s + 0.204·24-s − 1/5·25-s − 0.962·27-s + 2.22·29-s − 0.547·30-s + 0.176·32-s − 1/3·36-s − 0.324·38-s − 0.474·40-s + 1.06·43-s + 0.894·45-s + 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.323858007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.323858007\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 113 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−8.502107663437313837743360836102, −8.339203167091063860510120957967, −7.82417390518021578736974611077, −7.47840498287463594369107255975, −6.89176702131083233382962082909, −6.43092716332452152904063511592, −5.96409929697957297295485792826, −5.23901001816630187169322256311, −4.80886162074556748753824954145, −4.23578062232650500877146687790, −3.78674261373598701112600043432, −3.18444396654584534164663425197, −2.78201610350151334029445584968, −2.04397086816987791248313455789, −0.75153958703597358941007013711,
0.75153958703597358941007013711, 2.04397086816987791248313455789, 2.78201610350151334029445584968, 3.18444396654584534164663425197, 3.78674261373598701112600043432, 4.23578062232650500877146687790, 4.80886162074556748753824954145, 5.23901001816630187169322256311, 5.96409929697957297295485792826, 6.43092716332452152904063511592, 6.89176702131083233382962082909, 7.47840498287463594369107255975, 7.82417390518021578736974611077, 8.339203167091063860510120957967, 8.502107663437313837743360836102
