| L(s) = 1 | + 2-s − 2·3-s − 2·4-s − 2·6-s + 2·7-s − 3·8-s + 3·9-s − 4·11-s + 4·12-s + 2·13-s + 2·14-s + 16-s + 6·17-s + 3·18-s − 4·21-s − 4·22-s + 8·23-s + 6·24-s + 2·26-s − 4·27-s − 4·28-s − 2·29-s − 4·31-s + 2·32-s + 8·33-s + 6·34-s − 6·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s − 4-s − 0.816·6-s + 0.755·7-s − 1.06·8-s + 9-s − 1.20·11-s + 1.15·12-s + 0.554·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.707·18-s − 0.872·21-s − 0.852·22-s + 1.66·23-s + 1.22·24-s + 0.392·26-s − 0.769·27-s − 0.755·28-s − 0.371·29-s − 0.718·31-s + 0.353·32-s + 1.39·33-s + 1.02·34-s − 36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 107 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 33 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 2 T - 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 10 T + 102 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 50 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2 T + 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 101 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 170 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−7.40897391406383498406660535562, −7.23364478070541773066492035082, −7.16963634953140803486522381002, −6.53358138018195434591214884713, −5.93019035014758498409729296383, −5.82693659379033111013275057945, −5.37312702174188912675088903521, −5.28981632039806019278882603763, −4.86665660378284655692390599575, −4.72583032638865709956046037708, −4.17089919596359270282037885620, −3.90534058451608448728005536771, −3.31157694363779819151537939095, −3.15494786998611626932950732536, −2.59807439228722807123824730793, −1.87201460800011726509751272612, −1.23800182648834983810266662841, −1.14447031277632205970170556774, 0, 0,
1.14447031277632205970170556774, 1.23800182648834983810266662841, 1.87201460800011726509751272612, 2.59807439228722807123824730793, 3.15494786998611626932950732536, 3.31157694363779819151537939095, 3.90534058451608448728005536771, 4.17089919596359270282037885620, 4.72583032638865709956046037708, 4.86665660378284655692390599575, 5.28981632039806019278882603763, 5.37312702174188912675088903521, 5.82693659379033111013275057945, 5.93019035014758498409729296383, 6.53358138018195434591214884713, 7.16963634953140803486522381002, 7.23364478070541773066492035082, 7.40897391406383498406660535562
