| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s − 4·5-s − 4·6-s + 4·8-s − 9-s − 8·10-s + 2·11-s − 6·12-s − 2·13-s + 8·15-s + 5·16-s − 4·17-s − 2·18-s − 4·19-s − 12·20-s + 4·22-s − 4·23-s − 8·24-s + 4·25-s − 4·26-s + 6·27-s − 6·29-s + 16·30-s − 8·31-s + 6·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.78·5-s − 1.63·6-s + 1.41·8-s − 1/3·9-s − 2.52·10-s + 0.603·11-s − 1.73·12-s − 0.554·13-s + 2.06·15-s + 5/4·16-s − 0.970·17-s − 0.471·18-s − 0.917·19-s − 2.68·20-s + 0.852·22-s − 0.834·23-s − 1.63·24-s + 4/5·25-s − 0.784·26-s + 1.15·27-s − 1.11·29-s + 2.92·30-s − 1.43·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 32 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 16 T + 136 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 96 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 14 T + 165 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 18 T + 195 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 14 T + 165 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 108 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 208 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 18 T + 221 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T - 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 122 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 187 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−10.06477862546213881074250399468, −9.117571181381448836660539112961, −8.647391055893001048566521837963, −8.509975251680844266118807433803, −7.74950471439085965865414281976, −7.46933604672738163711201981400, −6.94669093177188615185586851706, −6.75197710276730297124267104510, −6.08331219643602788103178146321, −5.85494737514462564543108724177, −5.24453842468061287533273340138, −4.92220897291724393041969388405, −4.40137796680275312007265807935, −4.00316208833923154090235521243, −3.42506786528039792244107635968, −3.35509225231927757396983707596, −2.16178137203075410890896305845, −1.78873006098545990279967493870, 0, 0,
1.78873006098545990279967493870, 2.16178137203075410890896305845, 3.35509225231927757396983707596, 3.42506786528039792244107635968, 4.00316208833923154090235521243, 4.40137796680275312007265807935, 4.92220897291724393041969388405, 5.24453842468061287533273340138, 5.85494737514462564543108724177, 6.08331219643602788103178146321, 6.75197710276730297124267104510, 6.94669093177188615185586851706, 7.46933604672738163711201981400, 7.74950471439085965865414281976, 8.509975251680844266118807433803, 8.647391055893001048566521837963, 9.117571181381448836660539112961, 10.06477862546213881074250399468
