| L(s) = 1 | − 2·2-s − 4·3-s + 2·4-s − 3·5-s + 8·6-s − 2·7-s + 8·9-s + 6·10-s − 2·11-s − 8·12-s + 4·14-s + 12·15-s − 4·16-s + 17-s − 16·18-s + 2·19-s − 6·20-s + 8·21-s + 4·22-s − 8·23-s + 5·25-s − 12·27-s − 4·28-s − 7·29-s − 24·30-s + 4·31-s + 8·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 4-s − 1.34·5-s + 3.26·6-s − 0.755·7-s + 8/3·9-s + 1.89·10-s − 0.603·11-s − 2.30·12-s + 1.06·14-s + 3.09·15-s − 16-s + 0.242·17-s − 3.77·18-s + 0.458·19-s − 1.34·20-s + 1.74·21-s + 0.852·22-s − 1.66·23-s + 25-s − 2.30·27-s − 0.755·28-s − 1.29·29-s − 4.38·30-s + 0.718·31-s + 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 932 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 932 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 233 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 19 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 49 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 7 T + 56 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 48 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 4 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 130 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 173 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−19.6149831097, −19.0281141033, −18.5786811519, −18.1375797652, −17.5634728633, −17.0698281180, −16.5119533552, −16.1648430320, −15.7600008083, −15.2385647659, −13.9149801233, −13.0212809395, −12.2597221499, −11.7780865656, −11.4045994789, −10.6773595755, −10.2356390999, −9.53324027887, −8.41240371243, −7.61759364211, −7.03239593769, −6.09676745482, −5.26846820096, −4.02949124485, 0,
4.02949124485, 5.26846820096, 6.09676745482, 7.03239593769, 7.61759364211, 8.41240371243, 9.53324027887, 10.2356390999, 10.6773595755, 11.4045994789, 11.7780865656, 12.2597221499, 13.0212809395, 13.9149801233, 15.2385647659, 15.7600008083, 16.1648430320, 16.5119533552, 17.0698281180, 17.5634728633, 18.1375797652, 18.5786811519, 19.0281141033, 19.6149831097
