| L(s) = 1 | − 3-s − 5-s + 3·9-s + 2·11-s − 8·13-s + 15-s + 6·19-s − 3·23-s − 8·27-s − 6·29-s − 2·33-s + 12·37-s + 8·39-s + 14·41-s − 18·43-s − 3·45-s + 6·53-s − 2·55-s − 6·57-s − 10·59-s + 5·61-s + 8·65-s − 11·67-s + 3·69-s − 20·71-s − 8·73-s − 6·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 9-s + 0.603·11-s − 2.21·13-s + 0.258·15-s + 1.37·19-s − 0.625·23-s − 1.53·27-s − 1.11·29-s − 0.348·33-s + 1.97·37-s + 1.28·39-s + 2.18·41-s − 2.74·43-s − 0.447·45-s + 0.824·53-s − 0.269·55-s − 0.794·57-s − 1.30·59-s + 0.640·61-s + 0.992·65-s − 1.34·67-s + 0.361·69-s − 2.37·71-s − 0.936·73-s − 0.675·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7985913982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7985913982\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 8 T - 9 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 6 T - 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 17 T + 200 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.704738507828606238765624889563, −9.244533303323341467054082582756, −8.642014564346220515520014263425, −7.896139070633674251882418481490, −7.72439572473809463411575063494, −7.37033755159232893172793045398, −7.25724552969849724084746020623, −6.68348294478882415228247449853, −6.12461488300044872172871879894, −5.82366969356590017387397967732, −5.35025823624114750546671489536, −4.83335997937364640788419283639, −4.59507761212462334650850989884, −3.94977614155180221790272179035, −3.85063913259806463374045667238, −2.87133031446484704671836476377, −2.67134640564900498057431619074, −1.77955254203218709510698288686, −1.35625031084951966351389565260, −0.34522657021726829050161345472,
0.34522657021726829050161345472, 1.35625031084951966351389565260, 1.77955254203218709510698288686, 2.67134640564900498057431619074, 2.87133031446484704671836476377, 3.85063913259806463374045667238, 3.94977614155180221790272179035, 4.59507761212462334650850989884, 4.83335997937364640788419283639, 5.35025823624114750546671489536, 5.82366969356590017387397967732, 6.12461488300044872172871879894, 6.68348294478882415228247449853, 7.25724552969849724084746020623, 7.37033755159232893172793045398, 7.72439572473809463411575063494, 7.896139070633674251882418481490, 8.642014564346220515520014263425, 9.244533303323341467054082582756, 9.704738507828606238765624889563
