L(s) = 1 | + 2-s − 5·3-s + 5-s − 5·6-s − 7-s + 15·9-s + 10-s + 11-s − 14-s − 5·15-s + 17-s + 15·18-s − 19-s + 5·21-s + 22-s − 35·27-s − 5·30-s − 5·33-s + 34-s − 35-s − 37-s − 38-s + 5·42-s − 43-s + 15·45-s − 5·51-s − 35·54-s + ⋯ |
L(s) = 1 | + 2-s − 5·3-s + 5-s − 5·6-s − 7-s + 15·9-s + 10-s + 11-s − 14-s − 5·15-s + 17-s + 15·18-s − 19-s + 5·21-s + 22-s − 35·27-s − 5·30-s − 5·33-s + 34-s − 35-s − 37-s − 38-s + 5·42-s − 43-s + 15·45-s − 5·51-s − 35·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 197^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 197^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1691454834\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1691454834\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_1$ | \( ( 1 + T )^{5} \) |
| 197 | $C_1$ | \( ( 1 + T )^{5} \) |
good | 2 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 5 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 7 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 11 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 17 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 19 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 37 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 43 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 61 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 71 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 89 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 97 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.66237059500660415356588547842, −6.30426624176626154788582360319, −6.22552313158709465908198716243, −6.19623899487199539124863349461, −6.12329184503179824230534581395, −5.76896124072112502200366667927, −5.43701637745393464281762277777, −5.39087448232256428587576153025, −5.34653707680592805465235910286, −5.13051133890941151135768396585, −4.89096287639849296897434723588, −4.54135777902142320172860263600, −4.50655861583626393508981776065, −4.28591092828602881799368338735, −4.13106980047591615744440387510, −3.92019783737823609810945523924, −3.47580961333898468040158924752, −3.43098296892257340637838362892, −3.21192069259822105248431137681, −2.27185101973434999401567118730, −2.24500824390988576284965548980, −1.71661130588285073503029844428, −1.42980776969111633705730394341, −1.33022535577687141478853731364, −0.65344816078370669748145330666,
0.65344816078370669748145330666, 1.33022535577687141478853731364, 1.42980776969111633705730394341, 1.71661130588285073503029844428, 2.24500824390988576284965548980, 2.27185101973434999401567118730, 3.21192069259822105248431137681, 3.43098296892257340637838362892, 3.47580961333898468040158924752, 3.92019783737823609810945523924, 4.13106980047591615744440387510, 4.28591092828602881799368338735, 4.50655861583626393508981776065, 4.54135777902142320172860263600, 4.89096287639849296897434723588, 5.13051133890941151135768396585, 5.34653707680592805465235910286, 5.39087448232256428587576153025, 5.43701637745393464281762277777, 5.76896124072112502200366667927, 6.12329184503179824230534581395, 6.19623899487199539124863349461, 6.22552313158709465908198716243, 6.30426624176626154788582360319, 6.66237059500660415356588547842