| L(s) = 1 | + 2·3-s − 2·7-s + 3·9-s − 2·11-s − 2·13-s − 4·17-s − 2·19-s − 4·21-s + 4·23-s + 4·27-s − 2·29-s + 8·31-s − 4·33-s − 14·37-s − 4·39-s + 6·41-s − 14·43-s − 12·47-s + 2·49-s − 8·51-s − 4·57-s − 4·59-s − 6·63-s − 8·67-s + 8·69-s − 4·71-s − 12·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.755·7-s + 9-s − 0.603·11-s − 0.554·13-s − 0.970·17-s − 0.458·19-s − 0.872·21-s + 0.834·23-s + 0.769·27-s − 0.371·29-s + 1.43·31-s − 0.696·33-s − 2.30·37-s − 0.640·39-s + 0.937·41-s − 2.13·43-s − 1.75·47-s + 2/7·49-s − 1.12·51-s − 0.529·57-s − 0.520·59-s − 0.755·63-s − 0.977·67-s + 0.963·69-s − 0.474·71-s − 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 14 T + 110 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 14 T + 122 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 94 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 130 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 174 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 26 T + 350 T^{2} + 26 p T^{3} + p^{2} T^{4} \) |
| 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
−8.005496798035787714169369289047, −7.76345229110365980809229011535, −7.21631094202140810468760249785, −6.81835406276135499316239356856, −6.57260012380738724875131887615, −6.50850667230466942289921119201, −5.81738185366415115701398223682, −5.29336658304710231161040098333, −4.92209435464879378526023176096, −4.74826360047143999302929502166, −4.22655493173202345481604399752, −3.67975114986920961284829504326, −3.46550932879695626864448987419, −2.97138795116333100930575535638, −2.48538240482021548289882044039, −2.46177021673501135857725024930, −1.53899881658505912435319864963, −1.38604647597786512699173180083, 0, 0,
1.38604647597786512699173180083, 1.53899881658505912435319864963, 2.46177021673501135857725024930, 2.48538240482021548289882044039, 2.97138795116333100930575535638, 3.46550932879695626864448987419, 3.67975114986920961284829504326, 4.22655493173202345481604399752, 4.74826360047143999302929502166, 4.92209435464879378526023176096, 5.29336658304710231161040098333, 5.81738185366415115701398223682, 6.50850667230466942289921119201, 6.57260012380738724875131887615, 6.81835406276135499316239356856, 7.21631094202140810468760249785, 7.76345229110365980809229011535, 8.005496798035787714169369289047
