| L(s) = 1 | + 2·2-s − 3-s + 2·4-s + 5-s − 2·6-s + 7-s − 9-s + 2·10-s + 4·11-s − 2·12-s + 4·13-s + 2·14-s − 15-s − 4·16-s − 2·18-s − 4·19-s + 2·20-s − 21-s + 8·22-s + 7·23-s + 8·26-s + 2·28-s + 4·29-s − 2·30-s − 2·31-s − 8·32-s − 4·33-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 0.377·7-s − 1/3·9-s + 0.632·10-s + 1.20·11-s − 0.577·12-s + 1.10·13-s + 0.534·14-s − 0.258·15-s − 16-s − 0.471·18-s − 0.917·19-s + 0.447·20-s − 0.218·21-s + 1.70·22-s + 1.45·23-s + 1.56·26-s + 0.377·28-s + 0.742·29-s − 0.365·30-s − 0.359·31-s − 1.41·32-s − 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.922328676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.922328676\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T - 23 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 41 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 97 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 - 9 T + 96 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 146 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 14 T + 173 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 51 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + T - 8 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T - 18 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−14.2324966315, −13.8801198290, −13.4377149291, −12.9830221029, −12.6235785829, −12.1134293124, −11.6403506974, −11.3810713323, −10.7774659714, −10.6528536998, −9.74660330163, −9.16024522142, −8.75542821322, −8.44912054189, −7.51233710322, −6.73477375586, −6.55539742272, −6.04506772092, −5.37035096961, −5.19564209000, −4.16709076113, −4.02684394073, −3.15288425601, −2.35382354093, −1.26039399631,
1.26039399631, 2.35382354093, 3.15288425601, 4.02684394073, 4.16709076113, 5.19564209000, 5.37035096961, 6.04506772092, 6.55539742272, 6.73477375586, 7.51233710322, 8.44912054189, 8.75542821322, 9.16024522142, 9.74660330163, 10.6528536998, 10.7774659714, 11.3810713323, 11.6403506974, 12.1134293124, 12.6235785829, 12.9830221029, 13.4377149291, 13.8801198290, 14.2324966315
