| L(s) = 1 | + 3-s − 2·5-s − 7-s − 9-s + 4·11-s + 3·13-s − 2·15-s + 17-s + 12·19-s − 21-s − 2·23-s + 3·25-s − 4·29-s + 2·31-s + 4·33-s + 2·35-s + 5·37-s + 3·39-s + 2·45-s + 3·47-s − 9·49-s + 51-s + 5·53-s − 8·55-s + 12·57-s + 5·59-s + 6·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.377·7-s − 1/3·9-s + 1.20·11-s + 0.832·13-s − 0.516·15-s + 0.242·17-s + 2.75·19-s − 0.218·21-s − 0.417·23-s + 3/5·25-s − 0.742·29-s + 0.359·31-s + 0.696·33-s + 0.338·35-s + 0.821·37-s + 0.480·39-s + 0.298·45-s + 0.437·47-s − 9/7·49-s + 0.140·51-s + 0.686·53-s − 1.07·55-s + 1.58·57-s + 0.650·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.753706778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.753706778\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 24 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T - 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 76 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 65 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 3 T + 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 108 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 5 T + 86 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 114 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 13 T + 172 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 161 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 13 T + 184 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 18 T + 222 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 146 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 18 T + 258 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
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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
−7.958809338087667642792578460277, −7.70220661276333724748555838811, −7.43556330077570435237561720036, −7.18857883733054763352252444631, −6.78775237777701544694930213278, −6.34802904922270540575691617768, −5.83541372291831871211715516462, −5.82267525804955039661466265779, −5.34842455388611028795916728973, −4.79787925420876286180639712433, −4.26894993430780294019435712814, −4.24288408199445087644334357666, −3.51565355689779429989176572570, −3.40592808833750792815874636475, −3.00168493817362627186721164210, −2.82408753881766131791588886895, −1.92892183602620357217328160573, −1.50627063960727697358715620309, −0.977769859390679852269078789549, −0.54169251609379316456089322902,
0.54169251609379316456089322902, 0.977769859390679852269078789549, 1.50627063960727697358715620309, 1.92892183602620357217328160573, 2.82408753881766131791588886895, 3.00168493817362627186721164210, 3.40592808833750792815874636475, 3.51565355689779429989176572570, 4.24288408199445087644334357666, 4.26894993430780294019435712814, 4.79787925420876286180639712433, 5.34842455388611028795916728973, 5.82267525804955039661466265779, 5.83541372291831871211715516462, 6.34802904922270540575691617768, 6.78775237777701544694930213278, 7.18857883733054763352252444631, 7.43556330077570435237561720036, 7.70220661276333724748555838811, 7.958809338087667642792578460277
