| L(s) = 1 | + 2·2-s + 2·4-s + 6·7-s + 4·8-s − 8·11-s + 4·13-s + 12·14-s + 8·16-s + 2·17-s + 2·19-s − 16·22-s + 8·26-s + 12·28-s − 4·29-s − 6·31-s + 8·32-s + 4·34-s + 2·37-s + 4·38-s − 4·41-s + 10·43-s − 16·44-s + 14·47-s + 16·49-s + 8·52-s − 10·53-s + 24·56-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 2.26·7-s + 1.41·8-s − 2.41·11-s + 1.10·13-s + 3.20·14-s + 2·16-s + 0.485·17-s + 0.458·19-s − 3.41·22-s + 1.56·26-s + 2.26·28-s − 0.742·29-s − 1.07·31-s + 1.41·32-s + 0.685·34-s + 0.328·37-s + 0.648·38-s − 0.624·41-s + 1.52·43-s − 2.41·44-s + 2.04·47-s + 16/7·49-s + 1.10·52-s − 1.37·53-s + 3.20·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.207264415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.207264415\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 8 T + 35 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 32 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 35 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 72 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 59 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 84 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 14 T + 140 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 128 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 20 T + 215 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 143 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 72 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 24 T + 290 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 151 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 120 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.065754772786711019387360760225, −9.040314410663308323679746836517, −8.246885561424539140351888020424, −7.974675881482129984367119017506, −7.76192416674552620936952223598, −7.43434572855374939870549393441, −7.31985422575701095016963349574, −6.33220828587428272848186143010, −5.86961609715870612280308954431, −5.59136789897443639248731596957, −5.09479148926456285299721017379, −5.05046732407156589214777319240, −4.59200519527533780818264604353, −4.16946610424707768476471388278, −3.61647630049747472535524566854, −3.21245120689521848055503110768, −2.50830666356336420361556638435, −1.98245530703763601898170584253, −1.59915097235689293759400643519, −0.839772209097735318779632296013,
0.839772209097735318779632296013, 1.59915097235689293759400643519, 1.98245530703763601898170584253, 2.50830666356336420361556638435, 3.21245120689521848055503110768, 3.61647630049747472535524566854, 4.16946610424707768476471388278, 4.59200519527533780818264604353, 5.05046732407156589214777319240, 5.09479148926456285299721017379, 5.59136789897443639248731596957, 5.86961609715870612280308954431, 6.33220828587428272848186143010, 7.31985422575701095016963349574, 7.43434572855374939870549393441, 7.76192416674552620936952223598, 7.974675881482129984367119017506, 8.246885561424539140351888020424, 9.040314410663308323679746836517, 9.065754772786711019387360760225
