| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 4·6-s − 4·8-s + 3·9-s + 2·11-s + 6·12-s + 5·16-s − 8·17-s − 6·18-s − 8·19-s − 4·22-s + 4·23-s − 8·24-s − 10·25-s + 4·27-s − 8·31-s − 6·32-s + 4·33-s + 16·34-s + 9·36-s − 4·37-s + 16·38-s − 8·41-s − 20·43-s + 6·44-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.63·6-s − 1.41·8-s + 9-s + 0.603·11-s + 1.73·12-s + 5/4·16-s − 1.94·17-s − 1.41·18-s − 1.83·19-s − 0.852·22-s + 0.834·23-s − 1.63·24-s − 2·25-s + 0.769·27-s − 1.43·31-s − 1.06·32-s + 0.696·33-s + 2.74·34-s + 3/2·36-s − 0.657·37-s + 2.59·38-s − 1.24·41-s − 3.04·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10458756 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10458756 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 52 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 28 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 92 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 120 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 180 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 192 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
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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
−8.546207677958704399328037653021, −8.370126173621390129695589999820, −7.70647970725627220271923738030, −7.66515287981542189023396363562, −6.89803630701357608047630337438, −6.82555087595750569363787587804, −6.39464694160007385587922893641, −6.28201946977882066944995791577, −5.29945299814021959004927834765, −5.17999560291857478239417795123, −4.30919506497167881380120618170, −4.15363615994975899704543567273, −3.48880534897610477124931807098, −3.35243553816755163813653302649, −2.38330601354905729824208190344, −2.29830846019740276361077734996, −1.61175161727465132907500839358, −1.57238200269212342711657703280, 0, 0,
1.57238200269212342711657703280, 1.61175161727465132907500839358, 2.29830846019740276361077734996, 2.38330601354905729824208190344, 3.35243553816755163813653302649, 3.48880534897610477124931807098, 4.15363615994975899704543567273, 4.30919506497167881380120618170, 5.17999560291857478239417795123, 5.29945299814021959004927834765, 6.28201946977882066944995791577, 6.39464694160007385587922893641, 6.82555087595750569363787587804, 6.89803630701357608047630337438, 7.66515287981542189023396363562, 7.70647970725627220271923738030, 8.370126173621390129695589999820, 8.546207677958704399328037653021
