| L(s) = 1 | − 2·3-s + 2·4-s + 8·7-s + 3·9-s + 6·11-s − 4·12-s + 2·13-s + 6·17-s − 7·19-s − 16·21-s − 10·27-s + 16·28-s + 3·29-s − 14·31-s − 12·33-s + 6·36-s − 16·37-s − 4·39-s + 6·41-s − 4·43-s + 12·44-s + 6·47-s + 34·49-s − 12·51-s + 4·52-s − 6·53-s + 14·57-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 4-s + 3.02·7-s + 9-s + 1.80·11-s − 1.15·12-s + 0.554·13-s + 1.45·17-s − 1.60·19-s − 3.49·21-s − 1.92·27-s + 3.02·28-s + 0.557·29-s − 2.51·31-s − 2.08·33-s + 36-s − 2.63·37-s − 0.640·39-s + 0.937·41-s − 0.609·43-s + 1.80·44-s + 0.875·47-s + 34/7·49-s − 1.68·51-s + 0.554·52-s − 0.824·53-s + 1.85·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.496192694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.496192694\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 15 T + 166 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 3 T - 62 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 8 T - 9 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 8 T - 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.25680112181617328687651782649, −10.86667897450936964890459142585, −10.71593279964040892144337013764, −10.23460724456847848843181183484, −9.412174090981647598879264232348, −8.812319320166885670974834814309, −8.614350793160160309967491943006, −7.88098328543309962338861424988, −7.60332834659405498782401041705, −6.99493743745264066898582476958, −6.72053709103147934181714728347, −6.03635348207729154368808649323, −5.45898555686003375075929085438, −5.30231486765980986190015992088, −4.53081813332288507663557053442, −4.03663261716282539335541559312, −3.59627781104575035677850307993, −1.90450901963427336876409659814, −1.86919817563922856821254525608, −1.24096142123401968690163984206,
1.24096142123401968690163984206, 1.86919817563922856821254525608, 1.90450901963427336876409659814, 3.59627781104575035677850307993, 4.03663261716282539335541559312, 4.53081813332288507663557053442, 5.30231486765980986190015992088, 5.45898555686003375075929085438, 6.03635348207729154368808649323, 6.72053709103147934181714728347, 6.99493743745264066898582476958, 7.60332834659405498782401041705, 7.88098328543309962338861424988, 8.614350793160160309967491943006, 8.812319320166885670974834814309, 9.412174090981647598879264232348, 10.23460724456847848843181183484, 10.71593279964040892144337013764, 10.86667897450936964890459142585, 11.25680112181617328687651782649
