| L(s) = 1 | + 2-s − 5-s + 9-s − 10-s − 13-s + 18-s − 26-s + 2·29-s − 32-s − 3·37-s − 45-s + 4·49-s − 5·53-s + 2·58-s + 2·61-s − 64-s + 65-s + 2·73-s − 3·74-s − 5·89-s − 90-s − 2·97-s + 4·98-s − 2·101-s − 5·106-s − 5·113-s − 117-s + ⋯ |
| L(s) = 1 | + 2-s − 5-s + 9-s − 10-s − 13-s + 18-s − 26-s + 2·29-s − 32-s − 3·37-s − 45-s + 4·49-s − 5·53-s + 2·58-s + 2·61-s − 64-s + 65-s + 2·73-s − 3·74-s − 5·89-s − 90-s − 2·97-s + 4·98-s − 2·101-s − 5·106-s − 5·113-s − 117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.026128989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.026128989\) |
| \(L(1)\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 5 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 13 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| good | 3 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 11 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 17 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 19 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 23 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 29 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 31 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 37 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 41 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 53 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 59 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 61 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 67 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 71 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 73 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 79 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 89 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 97 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
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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
−7.11771675487351229396692243274, −6.76160533436640586893668077705, −6.67848716506550657474779882894, −6.60507829949639870985158534854, −6.45388441945623399627814550131, −5.84699987285958321368995002501, −5.48338217199191172364336052898, −5.45136365759392557317200524535, −5.43817507661809815025805033895, −5.04737195388509812474018239632, −4.80364133648008745912642108083, −4.63121880377543023509445274656, −4.37113472937462477961690003686, −4.02274640985583990502053933316, −4.00279435746061860784135134892, −3.84741738045711040326346702655, −3.62417435187499690843348748564, −3.09617288005201587909402820999, −2.81693272722253792575579381372, −2.62239382316628216093577842622, −2.58787711412689116049285053422, −1.72492685825827688896841110592, −1.56100148856249326080737018526, −1.50825976844226755048726003384, −0.56836032398823966226236385381,
0.56836032398823966226236385381, 1.50825976844226755048726003384, 1.56100148856249326080737018526, 1.72492685825827688896841110592, 2.58787711412689116049285053422, 2.62239382316628216093577842622, 2.81693272722253792575579381372, 3.09617288005201587909402820999, 3.62417435187499690843348748564, 3.84741738045711040326346702655, 4.00279435746061860784135134892, 4.02274640985583990502053933316, 4.37113472937462477961690003686, 4.63121880377543023509445274656, 4.80364133648008745912642108083, 5.04737195388509812474018239632, 5.43817507661809815025805033895, 5.45136365759392557317200524535, 5.48338217199191172364336052898, 5.84699987285958321368995002501, 6.45388441945623399627814550131, 6.60507829949639870985158534854, 6.67848716506550657474779882894, 6.76160533436640586893668077705, 7.11771675487351229396692243274
