| L(s) = 1 | + 2-s − 2·4-s − 7-s − 3·8-s + 11-s + 2·13-s − 14-s + 16-s + 6·17-s + 5·19-s + 22-s + 3·23-s + 2·26-s + 2·28-s + 10·29-s + 4·31-s + 2·32-s + 6·34-s − 6·37-s + 5·38-s − 4·41-s − 8·43-s − 2·44-s + 3·46-s + 21·47-s − 2·49-s − 4·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 4-s − 0.377·7-s − 1.06·8-s + 0.301·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 1.14·19-s + 0.213·22-s + 0.625·23-s + 0.392·26-s + 0.377·28-s + 1.85·29-s + 0.718·31-s + 0.353·32-s + 1.02·34-s − 0.986·37-s + 0.811·38-s − 0.624·41-s − 1.21·43-s − 0.301·44-s + 0.442·46-s + 3.06·47-s − 2/7·49-s − 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.067132095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.067132095\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 - T + 21 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 33 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 37 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 21 T + 203 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 63 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 141 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 14 T + 163 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 9 T + 101 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 7 T + 147 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 113 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 5 T + 183 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 19 T + 283 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
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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
−8.282958720866953813612542633455, −8.094460761003930993945612807778, −7.61557789765186445543412674660, −7.02823113912885768981149349096, −7.00181579344970994288328522218, −6.52430123915912970583070711053, −5.92560195941562975325390242219, −5.79966386556639881549127846467, −5.34716032522361109724944708762, −4.99858743636073959237800215998, −4.79609904800077239192706836926, −4.28518592567632890734254892747, −3.72340352006803077797176614669, −3.67740308093007128973072880150, −3.14775920986645090578136159743, −2.86357382756349243341096351945, −2.25628900022842097066028182388, −1.46741265662188887033377132700, −0.877761115581668772881106566872, −0.65784639508739265434806113932,
0.65784639508739265434806113932, 0.877761115581668772881106566872, 1.46741265662188887033377132700, 2.25628900022842097066028182388, 2.86357382756349243341096351945, 3.14775920986645090578136159743, 3.67740308093007128973072880150, 3.72340352006803077797176614669, 4.28518592567632890734254892747, 4.79609904800077239192706836926, 4.99858743636073959237800215998, 5.34716032522361109724944708762, 5.79966386556639881549127846467, 5.92560195941562975325390242219, 6.52430123915912970583070711053, 7.00181579344970994288328522218, 7.02823113912885768981149349096, 7.61557789765186445543412674660, 8.094460761003930993945612807778, 8.282958720866953813612542633455
