| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s − 2·5-s + 4·6-s − 2·7-s − 4·8-s + 3·9-s + 4·10-s − 2·11-s − 6·12-s + 2·13-s + 4·14-s + 4·15-s + 5·16-s − 2·17-s − 6·18-s − 2·19-s − 6·20-s + 4·21-s + 4·22-s + 2·23-s + 8·24-s + 3·25-s − 4·26-s − 4·27-s − 6·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s + 1.26·10-s − 0.603·11-s − 1.73·12-s + 0.554·13-s + 1.06·14-s + 1.03·15-s + 5/4·16-s − 0.485·17-s − 1.41·18-s − 0.458·19-s − 1.34·20-s + 0.872·21-s + 0.852·22-s + 0.417·23-s + 1.63·24-s + 3/5·25-s − 0.784·26-s − 0.769·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_4$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_4$ | \( 1 + 10 T + 114 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 18 T + 206 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 138 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 14 T + 190 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_4$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
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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
−7.72579528193371846491705213349, −7.69282559511759486298463732786, −7.42742845960879789994159223270, −6.93941952073029132568959007777, −6.52161677746673874930792949851, −6.32504887213968881735474383656, −5.95847921420948554713859963582, −5.72001551675743498886300221038, −4.98629576348545896343190146445, −4.85278165459810449003758763802, −4.09793630490715019314108975147, −4.06853825893635969694119455988, −3.31958827164474444638377309570, −2.97532646919772594957301820161, −2.53261331735217934787494582682, −1.92339056725753052652835142211, −1.30233501805382164955410724687, −0.849150571873832115931609819761, 0, 0,
0.849150571873832115931609819761, 1.30233501805382164955410724687, 1.92339056725753052652835142211, 2.53261331735217934787494582682, 2.97532646919772594957301820161, 3.31958827164474444638377309570, 4.06853825893635969694119455988, 4.09793630490715019314108975147, 4.85278165459810449003758763802, 4.98629576348545896343190146445, 5.72001551675743498886300221038, 5.95847921420948554713859963582, 6.32504887213968881735474383656, 6.52161677746673874930792949851, 6.93941952073029132568959007777, 7.42742845960879789994159223270, 7.69282559511759486298463732786, 7.72579528193371846491705213349
