L(s) = 1 | − 2-s − 3·3-s − 3·5-s + 3·6-s + 8-s + 6·9-s + 3·10-s + 3·11-s + 5·13-s + 9·15-s − 16-s − 6·17-s − 6·18-s − 10·19-s − 3·22-s + 3·23-s − 3·24-s + 5·25-s − 5·26-s − 9·27-s + 3·29-s − 9·30-s − 4·31-s − 9·33-s + 6·34-s − 14·37-s + 10·38-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.73·3-s − 1.34·5-s + 1.22·6-s + 0.353·8-s + 2·9-s + 0.948·10-s + 0.904·11-s + 1.38·13-s + 2.32·15-s − 1/4·16-s − 1.45·17-s − 1.41·18-s − 2.29·19-s − 0.639·22-s + 0.625·23-s − 0.612·24-s + 25-s − 0.980·26-s − 1.73·27-s + 0.557·29-s − 1.64·30-s − 0.718·31-s − 1.56·33-s + 1.02·34-s − 2.30·37-s + 1.62·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 3 T - 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−10.19562869270034713479775873560, −9.523756629809606465296259622115, −8.794416419967053015032500806870, −8.724726512086160746890719913753, −8.406572554738207186652410747970, −7.973657625687070944670092759898, −7.00337385015735356382566383571, −6.88102877667591105738239412144, −6.63130700951211029382758832697, −6.31161855661158165100669513573, −5.40240999221953605980898689631, −5.20092372061562563148816864730, −4.40176178283255397633132059669, −4.08761638957052283837292033521, −3.90816121421174497149503325854, −3.01586743033391632798713887408, −1.73913013012394302611000660936, −1.38710966815627223188826068273, 0, 0,
1.38710966815627223188826068273, 1.73913013012394302611000660936, 3.01586743033391632798713887408, 3.90816121421174497149503325854, 4.08761638957052283837292033521, 4.40176178283255397633132059669, 5.20092372061562563148816864730, 5.40240999221953605980898689631, 6.31161855661158165100669513573, 6.63130700951211029382758832697, 6.88102877667591105738239412144, 7.00337385015735356382566383571, 7.973657625687070944670092759898, 8.406572554738207186652410747970, 8.724726512086160746890719913753, 8.794416419967053015032500806870, 9.523756629809606465296259622115, 10.19562869270034713479775873560