| L(s) = 1 | − 3-s − 2·4-s − 2·5-s − 3·7-s − 2·9-s − 2·11-s + 2·12-s + 13-s + 2·15-s + 4·16-s − 5·17-s − 8·19-s + 4·20-s + 3·21-s − 23-s + 2·27-s + 6·28-s − 3·29-s − 7·31-s + 2·33-s + 6·35-s + 4·36-s − 6·37-s − 39-s − 4·41-s + 4·44-s + 4·45-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.894·5-s − 1.13·7-s − 2/3·9-s − 0.603·11-s + 0.577·12-s + 0.277·13-s + 0.516·15-s + 16-s − 1.21·17-s − 1.83·19-s + 0.894·20-s + 0.654·21-s − 0.208·23-s + 0.384·27-s + 1.13·28-s − 0.557·29-s − 1.25·31-s + 0.348·33-s + 1.01·35-s + 2/3·36-s − 0.986·37-s − 0.160·39-s − 0.624·41-s + 0.603·44-s + 0.596·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3637 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 22 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 3 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - T + 8 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T + 13 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7 T + 46 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + 4 T - 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 72 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 11 T + 55 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 7 T - 3 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 + 3 T + 26 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 5 T + 10 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 10 T + 80 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - T + 52 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−14.4473179436, −13.7902167518, −13.2564777809, −13.0936812300, −12.6857125394, −12.2419212793, −11.8074799595, −11.2036879198, −10.8216623022, −10.5152002135, −9.98506358419, −9.27865255927, −9.02110277025, −8.39670830023, −8.29851606830, −7.42716783818, −7.01367337735, −6.29461300009, −5.95976119474, −5.37807159995, −4.73314664729, −4.12109501492, −3.70235521265, −3.03187020814, −2.03967890047, 0, 0,
2.03967890047, 3.03187020814, 3.70235521265, 4.12109501492, 4.73314664729, 5.37807159995, 5.95976119474, 6.29461300009, 7.01367337735, 7.42716783818, 8.29851606830, 8.39670830023, 9.02110277025, 9.27865255927, 9.98506358419, 10.5152002135, 10.8216623022, 11.2036879198, 11.8074799595, 12.2419212793, 12.6857125394, 13.0936812300, 13.2564777809, 13.7902167518, 14.4473179436
