| L(s) = 1 | − 2-s − 3·3-s − 2·4-s + 3·6-s − 6·7-s + 3·8-s + 2·9-s − 11-s + 6·12-s + 2·13-s + 6·14-s + 16-s − 6·17-s − 2·18-s + 18·21-s + 22-s − 13·23-s − 9·24-s − 2·26-s + 6·27-s + 12·28-s − 5·29-s − 11·31-s − 2·32-s + 3·33-s + 6·34-s − 4·36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 4-s + 1.22·6-s − 2.26·7-s + 1.06·8-s + 2/3·9-s − 0.301·11-s + 1.73·12-s + 0.554·13-s + 1.60·14-s + 1/4·16-s − 1.45·17-s − 0.471·18-s + 3.92·21-s + 0.213·22-s − 2.71·23-s − 1.83·24-s − 0.392·26-s + 1.15·27-s + 2.26·28-s − 0.928·29-s − 1.97·31-s − 0.353·32-s + 0.522·33-s + 1.02·34-s − 2/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | | \( 1 \) |
| 19 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_4$ | \( 1 + T + 21 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 13 T + 87 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 63 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 11 T + 81 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 11 T + 93 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 7 T + 87 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 47 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 15 T + 113 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 16 T + 181 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 131 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 117 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 162 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 20 T + 273 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 21 T + 293 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 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
−7.10291334505861014976348259278, −6.63499114485165053461344075340, −6.58108024432842240100928909302, −6.19113353714644464931651932211, −5.81207217659286280836400848484, −5.69495890711204515545351744512, −5.23212234624897575972634101903, −4.98139648877440494518488031172, −4.33240300002455137954587219552, −4.06612607677978245676459272664, −3.77138959132062611814378260721, −3.40283148983620936945769696008, −2.83510260492345328558137922831, −2.45509737793981009181570837059, −1.54156526200735277543915975528, −1.49185935022862112481008762478, 0, 0, 0, 0,
1.49185935022862112481008762478, 1.54156526200735277543915975528, 2.45509737793981009181570837059, 2.83510260492345328558137922831, 3.40283148983620936945769696008, 3.77138959132062611814378260721, 4.06612607677978245676459272664, 4.33240300002455137954587219552, 4.98139648877440494518488031172, 5.23212234624897575972634101903, 5.69495890711204515545351744512, 5.81207217659286280836400848484, 6.19113353714644464931651932211, 6.58108024432842240100928909302, 6.63499114485165053461344075340, 7.10291334505861014976348259278
