| L(s) = 1 | − 2·2-s + 3·4-s − 3·5-s − 2·7-s − 4·8-s + 6·10-s − 11-s + 2·13-s + 4·14-s + 5·16-s + 17-s + 3·19-s − 9·20-s + 2·22-s + 7·23-s + 25-s − 4·26-s − 6·28-s − 29-s − 4·31-s − 6·32-s − 2·34-s + 6·35-s − 11·37-s − 6·38-s + 12·40-s − 6·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.34·5-s − 0.755·7-s − 1.41·8-s + 1.89·10-s − 0.301·11-s + 0.554·13-s + 1.06·14-s + 5/4·16-s + 0.242·17-s + 0.688·19-s − 2.01·20-s + 0.426·22-s + 1.45·23-s + 1/5·25-s − 0.784·26-s − 1.13·28-s − 0.185·29-s − 0.718·31-s − 1.06·32-s − 0.342·34-s + 1.01·35-s − 1.80·37-s − 0.973·38-s + 1.89·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 11 T + 100 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 17 T + 154 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - T + 84 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 126 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 3 T + 144 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 14 T + 190 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 26 T + 318 T^{2} + 26 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 4 T - 74 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 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
−8.943546469941654705387114337009, −8.786681186988976237867539731911, −8.368753807190748414751207446163, −8.269562579899544367306491342018, −7.46708607069936873467966307679, −7.33181853461224455582846520339, −6.89280920834416277493602414599, −6.82486306202951059930453674231, −5.90726272870591041979045530985, −5.75786970631544080979429890362, −5.04332455074417045923424864103, −4.69516632185330918124084363894, −3.67375410635739582237562205909, −3.67279003312917711289435593032, −3.06451183452278075127846203459, −2.72240667229055867481077864761, −1.60297949560182135315821130298, −1.33777036551990798132863711896, 0, 0,
1.33777036551990798132863711896, 1.60297949560182135315821130298, 2.72240667229055867481077864761, 3.06451183452278075127846203459, 3.67279003312917711289435593032, 3.67375410635739582237562205909, 4.69516632185330918124084363894, 5.04332455074417045923424864103, 5.75786970631544080979429890362, 5.90726272870591041979045530985, 6.82486306202951059930453674231, 6.89280920834416277493602414599, 7.33181853461224455582846520339, 7.46708607069936873467966307679, 8.269562579899544367306491342018, 8.368753807190748414751207446163, 8.786681186988976237867539731911, 8.943546469941654705387114337009
