| L(s) = 1 | − 2-s − 2·3-s + 2·6-s + 8-s + 3·9-s − 2·11-s − 16-s − 17-s − 3·18-s − 19-s + 2·22-s − 2·24-s + 2·25-s − 4·27-s + 4·33-s + 34-s + 38-s − 41-s + 43-s + 2·48-s − 2·50-s + 2·51-s + 4·54-s + 2·57-s − 59-s + 64-s − 4·66-s + ⋯ |
| L(s) = 1 | − 2-s − 2·3-s + 2·6-s + 8-s + 3·9-s − 2·11-s − 16-s − 17-s − 3·18-s − 19-s + 2·22-s − 2·24-s + 2·25-s − 4·27-s + 4·33-s + 34-s + 38-s − 41-s + 43-s + 2·48-s − 2·50-s + 2·51-s + 4·54-s + 2·57-s − 59-s + 64-s − 4·66-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1403040439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1403040439\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
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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
−8.960015034139428585056379756583, −8.616606714663420952236495605054, −8.179716970705391039201441908813, −7.66860064383304134040964046895, −7.53475555715532985510426433076, −7.16363099557469327309112437386, −6.61697613230895577108834359555, −6.33164101488568298723070583264, −6.20142796470599633613551456457, −5.28213576482073163866129744887, −5.21368660857939697154599063451, −4.87396443684704615000062290957, −4.66696738729661584203730275865, −3.99711758633983183236691015860, −3.73360089985589739853998253084, −2.67578245681792176034752873628, −2.42071744323730659207743186455, −1.70046031026273636474131648898, −1.12500370143614322942335816490, −0.34225332226001629267785543443,
0.34225332226001629267785543443, 1.12500370143614322942335816490, 1.70046031026273636474131648898, 2.42071744323730659207743186455, 2.67578245681792176034752873628, 3.73360089985589739853998253084, 3.99711758633983183236691015860, 4.66696738729661584203730275865, 4.87396443684704615000062290957, 5.21368660857939697154599063451, 5.28213576482073163866129744887, 6.20142796470599633613551456457, 6.33164101488568298723070583264, 6.61697613230895577108834359555, 7.16363099557469327309112437386, 7.53475555715532985510426433076, 7.66860064383304134040964046895, 8.179716970705391039201441908813, 8.616606714663420952236495605054, 8.960015034139428585056379756583
