L(s) = 1 | + 2·2-s + 2·3-s + 4-s − 2·5-s + 4·6-s − 2·7-s − 2·8-s + 9-s − 4·10-s + 12·11-s + 2·12-s − 6·13-s − 4·14-s − 4·15-s − 4·16-s − 12·17-s + 2·18-s − 2·20-s − 4·21-s + 24·22-s + 8·23-s − 4·24-s + 25-s − 12·26-s − 2·27-s − 2·28-s − 8·30-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 1.15·3-s + 1/2·4-s − 0.894·5-s + 1.63·6-s − 0.755·7-s − 0.707·8-s + 1/3·9-s − 1.26·10-s + 3.61·11-s + 0.577·12-s − 1.66·13-s − 1.06·14-s − 1.03·15-s − 16-s − 2.91·17-s + 0.471·18-s − 0.447·20-s − 0.872·21-s + 5.11·22-s + 1.66·23-s − 0.816·24-s + 1/5·25-s − 2.35·26-s − 0.384·27-s − 0.377·28-s − 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.840707830\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.840707830\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
good | 7 | $D_4\times C_2$ | \( 1 + 2 T - 8 T^{2} - 4 T^{3} + 67 T^{4} - 4 p T^{5} - 8 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 + 6 T + p T^{2} )^{2}( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $D_4\times C_2$ | \( 1 + 12 T + 77 T^{2} + 396 T^{3} + 1752 T^{4} + 396 p T^{5} + 77 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 23 | $D_{4}$ | \( ( 1 - 4 T + 2 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 18 T + 4 p T^{2} + 1404 T^{3} + 10635 T^{4} + 1404 p T^{5} + 4 p^{3} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 2 T + 83 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 10 T - 28 T^{2} - 220 T^{3} + 8275 T^{4} - 220 p T^{5} - 28 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 6 T - 79 T^{2} + 18 T^{3} + 7620 T^{4} + 18 p T^{5} - 79 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 6 T - 20 T^{2} + 396 T^{3} - 2781 T^{4} + 396 p T^{5} - 20 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 12 T - 23 T^{2} + 396 T^{3} + 13752 T^{4} + 396 p T^{5} - 23 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 10 T - 64 T^{2} - 220 T^{3} + 14755 T^{4} - 220 p T^{5} - 64 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 17 T + p T^{2} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 10 T - 16 T^{2} - 500 T^{3} - 1733 T^{4} - 500 p T^{5} - 16 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^3$ | \( 1 - 166 T^{2} + 19635 T^{4} - 166 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.94033243900043366850701057559, −6.87492421603630711497166676303, −6.71089782231517505158627638076, −6.33605326362960612212773878672, −6.30212803054232306825719665248, −5.93990648940510054628606418737, −5.73715361428010189329234988800, −5.24950951223589707062539607415, −5.17923520161089227997263395432, −4.88996496520490749239706477014, −4.61878976927427648601995398012, −4.32663463209397410598149125321, −4.22510594761302530400865871266, −3.87754848510148034441090222773, −3.82052921229067607445934233063, −3.79150546008028972037390924108, −3.33513711040180131429715732365, −2.96302484360948634203723740609, −2.78218723760458684382948552834, −2.45370402681156999716958558644, −2.25798691326153423309740886718, −1.94051119686729965121213572538, −1.18490993098610256117127859155, −1.17435237904192859844123331862, −0.20924664509479009760622936300,
0.20924664509479009760622936300, 1.17435237904192859844123331862, 1.18490993098610256117127859155, 1.94051119686729965121213572538, 2.25798691326153423309740886718, 2.45370402681156999716958558644, 2.78218723760458684382948552834, 2.96302484360948634203723740609, 3.33513711040180131429715732365, 3.79150546008028972037390924108, 3.82052921229067607445934233063, 3.87754848510148034441090222773, 4.22510594761302530400865871266, 4.32663463209397410598149125321, 4.61878976927427648601995398012, 4.88996496520490749239706477014, 5.17923520161089227997263395432, 5.24950951223589707062539607415, 5.73715361428010189329234988800, 5.93990648940510054628606418737, 6.30212803054232306825719665248, 6.33605326362960612212773878672, 6.71089782231517505158627638076, 6.87492421603630711497166676303, 6.94033243900043366850701057559