| L(s) = 1 | − 2·5-s + 2·7-s − 4·11-s − 2·13-s − 4·17-s − 2·19-s − 6·23-s − 4·25-s + 2·31-s − 4·35-s − 10·37-s − 2·41-s + 2·43-s − 8·49-s + 8·55-s − 4·59-s − 14·61-s + 4·65-s − 12·71-s − 6·73-s − 8·77-s − 12·79-s − 6·83-s + 8·85-s − 4·89-s − 4·91-s + 4·95-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s − 1.20·11-s − 0.554·13-s − 0.970·17-s − 0.458·19-s − 1.25·23-s − 4/5·25-s + 0.359·31-s − 0.676·35-s − 1.64·37-s − 0.312·41-s + 0.304·43-s − 8/7·49-s + 1.07·55-s − 0.520·59-s − 1.79·61-s + 0.496·65-s − 1.42·71-s − 0.702·73-s − 0.911·77-s − 1.35·79-s − 0.658·83-s + 0.867·85-s − 0.423·89-s − 0.419·91-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 51 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 91 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 151 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 107 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 146 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 172 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 10 T + 144 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−9.171652892180838274598613803997, −8.875584916088055041760577212192, −8.316652226279072360566313388847, −8.082472485893118240353996651706, −7.77100717687584969768338832487, −7.48827302553579699267360672094, −6.82442939511164237791252423881, −6.66359429740352601703748588023, −5.75917123832548301147292287641, −5.73105360976960927768519383055, −5.00966856544437803803215181895, −4.63930820306159591752185491693, −4.18426199824165909557514607678, −3.94827226633824101023975944983, −2.97971535932134946973597420543, −2.83443433405248140990662919405, −1.85483982575895652642671387053, −1.69514259255138328161664747353, 0, 0,
1.69514259255138328161664747353, 1.85483982575895652642671387053, 2.83443433405248140990662919405, 2.97971535932134946973597420543, 3.94827226633824101023975944983, 4.18426199824165909557514607678, 4.63930820306159591752185491693, 5.00966856544437803803215181895, 5.73105360976960927768519383055, 5.75917123832548301147292287641, 6.66359429740352601703748588023, 6.82442939511164237791252423881, 7.48827302553579699267360672094, 7.77100717687584969768338832487, 8.082472485893118240353996651706, 8.316652226279072360566313388847, 8.875584916088055041760577212192, 9.171652892180838274598613803997
