| L(s) = 1 | + 2-s + 2·3-s + 2·5-s + 2·6-s − 8-s + 3·9-s + 2·10-s − 11-s + 8·13-s + 4·15-s − 16-s + 3·18-s + 4·19-s − 22-s − 4·23-s − 2·24-s + 5·25-s + 8·26-s + 10·27-s + 4·29-s + 4·30-s − 10·31-s − 2·33-s + 6·37-s + 4·38-s + 16·39-s − 2·40-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 0.894·5-s + 0.816·6-s − 0.353·8-s + 9-s + 0.632·10-s − 0.301·11-s + 2.21·13-s + 1.03·15-s − 1/4·16-s + 0.707·18-s + 0.917·19-s − 0.213·22-s − 0.834·23-s − 0.408·24-s + 25-s + 1.56·26-s + 1.92·27-s + 0.742·29-s + 0.730·30-s − 1.79·31-s − 0.348·33-s + 0.986·37-s + 0.648·38-s + 2.56·39-s − 0.316·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.665019613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.665019613\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 10 T + 53 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−10.01350204916070635451877289179, −9.729518703746625427036580236451, −9.033285076876109913253813217025, −8.769125756712090139796204094030, −8.678046288002148475484347135518, −8.202460298631693284486675252214, −7.39213862833059705124936528491, −7.37919605047910310857874139410, −6.62899288222405712162424905873, −6.09765282408618968103522858111, −5.96189373856331395141331590868, −5.32632710212094652290359697579, −4.93056120075052739828509316368, −4.26311917124467220159868590220, −3.66539088643647559814240604776, −3.60901393889998111002004265296, −2.68805756499205694979615474203, −2.55196958815236995648887714010, −1.53096879039288438458703556727, −1.11876509928895296265418184908,
1.11876509928895296265418184908, 1.53096879039288438458703556727, 2.55196958815236995648887714010, 2.68805756499205694979615474203, 3.60901393889998111002004265296, 3.66539088643647559814240604776, 4.26311917124467220159868590220, 4.93056120075052739828509316368, 5.32632710212094652290359697579, 5.96189373856331395141331590868, 6.09765282408618968103522858111, 6.62899288222405712162424905873, 7.37919605047910310857874139410, 7.39213862833059705124936528491, 8.202460298631693284486675252214, 8.678046288002148475484347135518, 8.769125756712090139796204094030, 9.033285076876109913253813217025, 9.729518703746625427036580236451, 10.01350204916070635451877289179
