| L(s) = 1 | + 4·13-s + 2·17-s − 4·25-s − 2·29-s − 2·37-s + 2·41-s − 49-s + 2·61-s − 2·89-s + 2·97-s + 2·101-s + 2·113-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 4·13-s + 2·17-s − 4·25-s − 2·29-s − 2·37-s + 2·41-s − 49-s + 2·61-s − 2·89-s + 2·97-s + 2·101-s + 2·113-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.913611793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.913611793\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 7 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.25386161957017360222945412750, −5.85099254782685581030434711768, −5.78765406991829848564311505319, −5.71309792868014311543426564420, −5.59905036174912904897011108129, −5.41854634854879107883287409494, −5.01946736896274847563390753594, −4.88046544312526785997962206096, −4.65252383435788242339383980389, −4.09759718525409104929324204780, −3.97919822360119761812009064347, −3.95749065989775475071918108391, −3.84873173974687364641393048571, −3.48326567988762892935703111645, −3.42801944631849603077841054838, −3.36882483790819900265284104356, −3.00510465094881321589551489382, −2.60679838786445375692486019532, −2.08527953424548023632994253224, −2.07148267056603415708706039810, −1.95606871014051901303345007386, −1.39083069657033425645385176931, −1.25587135189913772951286599086, −1.19617245868019273480185179574, −0.50456235315438916296005531625,
0.50456235315438916296005531625, 1.19617245868019273480185179574, 1.25587135189913772951286599086, 1.39083069657033425645385176931, 1.95606871014051901303345007386, 2.07148267056603415708706039810, 2.08527953424548023632994253224, 2.60679838786445375692486019532, 3.00510465094881321589551489382, 3.36882483790819900265284104356, 3.42801944631849603077841054838, 3.48326567988762892935703111645, 3.84873173974687364641393048571, 3.95749065989775475071918108391, 3.97919822360119761812009064347, 4.09759718525409104929324204780, 4.65252383435788242339383980389, 4.88046544312526785997962206096, 5.01946736896274847563390753594, 5.41854634854879107883287409494, 5.59905036174912904897011108129, 5.71309792868014311543426564420, 5.78765406991829848564311505319, 5.85099254782685581030434711768, 6.25386161957017360222945412750
