| L(s) = 1 | + 2·5-s + 2·13-s − 2·17-s + 3·25-s + 2·37-s − 4·41-s + 2·53-s + 4·65-s − 2·73-s − 4·85-s − 2·97-s + 2·113-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 2·5-s + 2·13-s − 2·17-s + 3·25-s + 2·37-s − 4·41-s + 2·53-s + 4·65-s − 2·73-s − 4·85-s − 2·97-s + 2·113-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.731149706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.731149706\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2^2$ | \( 1 + T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 41 | $C_1$ | \( ( 1 + T )^{4} \) |
| 43 | $C_2^2$ | \( 1 + T^{4} \) |
| 47 | $C_2^2$ | \( 1 + T^{4} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + T^{4} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2^2$ | \( 1 + T^{4} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
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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
−9.750468171704631929740761426681, −9.704891110795053292508302904530, −8.958377295751747429461176436404, −8.819825118749431193299193415884, −8.379783667186916001121314056584, −8.356620180901545215757275275054, −7.22003887251406042531982736089, −7.04522918523971434102280963918, −6.50447129242142655992783928096, −6.19668732674116076915441533266, −5.99698087262425325432497810660, −5.47128246144532710457912472558, −4.91304121345842909654633126375, −4.63430813009946904863747690975, −3.87085004248764801643048610709, −3.51573612738716185265442853736, −2.59686168500782016158249376578, −2.45924310097564594588447621042, −1.57524917341929851110694534915, −1.32639248226309068763830315054,
1.32639248226309068763830315054, 1.57524917341929851110694534915, 2.45924310097564594588447621042, 2.59686168500782016158249376578, 3.51573612738716185265442853736, 3.87085004248764801643048610709, 4.63430813009946904863747690975, 4.91304121345842909654633126375, 5.47128246144532710457912472558, 5.99698087262425325432497810660, 6.19668732674116076915441533266, 6.50447129242142655992783928096, 7.04522918523971434102280963918, 7.22003887251406042531982736089, 8.356620180901545215757275275054, 8.379783667186916001121314056584, 8.819825118749431193299193415884, 8.958377295751747429461176436404, 9.704891110795053292508302904530, 9.750468171704631929740761426681
