| L(s) = 1 | + 2-s − 4·3-s − 2·4-s − 4·6-s − 6·7-s − 3·8-s + 6·9-s + 8·12-s − 7·13-s − 6·14-s + 16-s − 6·17-s + 6·18-s − 8·19-s + 24·21-s − 9·23-s + 12·24-s − 7·26-s + 4·27-s + 12·28-s − 12·29-s + 4·31-s + 2·32-s − 6·34-s − 12·36-s − 8·37-s − 8·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 2.30·3-s − 4-s − 1.63·6-s − 2.26·7-s − 1.06·8-s + 2·9-s + 2.30·12-s − 1.94·13-s − 1.60·14-s + 1/4·16-s − 1.45·17-s + 1.41·18-s − 1.83·19-s + 5.23·21-s − 1.87·23-s + 2.44·24-s − 1.37·26-s + 0.769·27-s + 2.26·28-s − 2.22·29-s + 0.718·31-s + 0.353·32-s − 1.02·34-s − 2·36-s − 1.31·37-s − 1.29·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43890625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43890625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 53 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 7 T + 37 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 49 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 9 T + 65 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 61 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 45 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 85 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 18 T + 170 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 9 T + 107 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 - 19 T + 201 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 7 T + 135 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 11 T + 171 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 26 T + 310 T^{2} + 26 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 14 T + 187 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 87 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + T + 177 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 11 T + 123 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
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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
−7.08447348646786401979462397378, −6.80680233800298807775071082339, −6.62257180226094632127499934572, −6.39815976920671717998557169477, −5.83817665932347132915016098611, −5.81253178148147372218102533053, −5.25631191530780437388139807876, −5.19474730322608963745321131981, −4.48685028225218570473488338691, −4.42633879832736563229386369687, −3.93080047353774947479247165407, −3.77034727732211643526188082049, −2.86910322907185950815828172531, −2.75907549635824004950488188758, −2.14778652738824503159865395919, −1.46745035877378628273348350851, 0, 0, 0, 0,
1.46745035877378628273348350851, 2.14778652738824503159865395919, 2.75907549635824004950488188758, 2.86910322907185950815828172531, 3.77034727732211643526188082049, 3.93080047353774947479247165407, 4.42633879832736563229386369687, 4.48685028225218570473488338691, 5.19474730322608963745321131981, 5.25631191530780437388139807876, 5.81253178148147372218102533053, 5.83817665932347132915016098611, 6.39815976920671717998557169477, 6.62257180226094632127499934572, 6.80680233800298807775071082339, 7.08447348646786401979462397378
