| L(s) = 1 | − 2·2-s − 3-s − 5-s + 2·6-s − 5·7-s + 4·8-s + 2·10-s − 2·11-s − 3·13-s + 10·14-s + 15-s − 4·16-s + 6·17-s − 3·19-s + 5·21-s + 4·22-s − 3·23-s − 4·24-s − 3·25-s + 6·26-s + 4·27-s − 2·30-s − 2·31-s + 2·33-s − 12·34-s + 5·35-s + 3·37-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s − 0.447·5-s + 0.816·6-s − 1.88·7-s + 1.41·8-s + 0.632·10-s − 0.603·11-s − 0.832·13-s + 2.67·14-s + 0.258·15-s − 16-s + 1.45·17-s − 0.688·19-s + 1.09·21-s + 0.852·22-s − 0.625·23-s − 0.816·24-s − 3/5·25-s + 1.17·26-s + 0.769·27-s − 0.365·30-s − 0.359·31-s + 0.348·33-s − 2.05·34-s + 0.845·35-s + 0.493·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2031 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2031 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 677 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 16 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 40 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + T + 19 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3 T - 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 + 7 T + 110 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 112 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 9 T + 154 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 134 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.8838819034, −18.6754163268, −18.1076996963, −17.5369349874, −16.9395293973, −16.5931756770, −16.2142617490, −15.6323979483, −14.8839879949, −14.1619639633, −13.4021768188, −12.9095633276, −12.2919268808, −11.8786113423, −10.7258527563, −10.2753906034, −9.68604757651, −9.48719337302, −8.51775231553, −7.90732979383, −7.24195555160, −6.32030536386, −5.49640245991, −4.35037126257, −3.14187462525, 0,
3.14187462525, 4.35037126257, 5.49640245991, 6.32030536386, 7.24195555160, 7.90732979383, 8.51775231553, 9.48719337302, 9.68604757651, 10.2753906034, 10.7258527563, 11.8786113423, 12.2919268808, 12.9095633276, 13.4021768188, 14.1619639633, 14.8839879949, 15.6323979483, 16.2142617490, 16.5931756770, 16.9395293973, 17.5369349874, 18.1076996963, 18.6754163268, 18.8838819034
