| L(s) = 1 | + 3-s − 2·4-s − 2·5-s + 5·7-s − 3·9-s − 4·11-s − 2·12-s − 3·13-s − 2·15-s + 4·16-s + 2·17-s + 4·20-s + 5·21-s + 4·23-s − 2·25-s − 4·27-s − 10·28-s − 4·29-s − 4·33-s − 10·35-s + 6·36-s − 3·39-s − 10·41-s + 4·43-s + 8·44-s + 6·45-s − 6·47-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.894·5-s + 1.88·7-s − 9-s − 1.20·11-s − 0.577·12-s − 0.832·13-s − 0.516·15-s + 16-s + 0.485·17-s + 0.894·20-s + 1.09·21-s + 0.834·23-s − 2/5·25-s − 0.769·27-s − 1.88·28-s − 0.742·29-s − 0.696·33-s − 1.69·35-s + 36-s − 0.480·39-s − 1.56·41-s + 0.609·43-s + 1.20·44-s + 0.894·45-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85360 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85360 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 3 T + p T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 3 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 33 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 10 T + 102 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 14 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 + T + 72 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 14 T + 151 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 9 T + 48 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 22 T + 234 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + T - 60 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 - 8 T - 19 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−14.4634203682, −14.1410695875, −13.4579279225, −13.3967500551, −12.6878635974, −12.1271841799, −11.7625676091, −11.3935164933, −10.8890292993, −10.4369771970, −9.90303340720, −9.27462404157, −8.68808219027, −8.50377025044, −7.89596779905, −7.63725717028, −7.46074495691, −6.23910461439, −5.45955842679, −4.98879084112, −4.86185252878, −4.01083513331, −3.31111221779, −2.66710303655, −1.63812281147, 0,
1.63812281147, 2.66710303655, 3.31111221779, 4.01083513331, 4.86185252878, 4.98879084112, 5.45955842679, 6.23910461439, 7.46074495691, 7.63725717028, 7.89596779905, 8.50377025044, 8.68808219027, 9.27462404157, 9.90303340720, 10.4369771970, 10.8890292993, 11.3935164933, 11.7625676091, 12.1271841799, 12.6878635974, 13.3967500551, 13.4579279225, 14.1410695875, 14.4634203682
