| L(s) = 1 | − 2-s − 5-s + 5·9-s + 10-s + 5·11-s − 13-s − 5·18-s − 5·22-s + 26-s − 31-s − 37-s − 41-s − 43-s − 5·45-s − 47-s + 5·49-s − 53-s − 5·55-s + 62-s + 65-s − 67-s + 74-s − 79-s + 15·81-s + 82-s − 83-s + 86-s + ⋯ |
| L(s) = 1 | − 2-s − 5-s + 5·9-s + 10-s + 5·11-s − 13-s − 5·18-s − 5·22-s + 26-s − 31-s − 37-s − 41-s − 43-s − 5·45-s − 47-s + 5·49-s − 53-s − 5·55-s + 62-s + 65-s − 67-s + 74-s − 79-s + 15·81-s + 82-s − 83-s + 86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{5} \cdot 149^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{5} \cdot 149^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.207538610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.207538610\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | $C_1$ | \( ( 1 - T )^{5} \) |
| 149 | $C_1$ | \( ( 1 - T )^{5} \) |
| good | 2 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 13 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 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 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 37 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 41 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 43 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 47 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 53 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 67 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 79 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 83 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
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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
−5.96718069782608620066582499746, −5.69417239012704625480313706885, −5.42800318493905304590272782572, −5.21562444954098450256128095981, −5.21488663269795335486138474278, −4.71227551592639260788695858267, −4.53865196170380346089488966297, −4.50789768954533815403045208375, −4.32872590968207422746806250725, −4.14127434968027192345883800648, −3.99660550496695633748179857107, −3.93851635189379161711132383221, −3.88901512924952851831933267317, −3.48217582897712719409779456911, −3.46308262692224247159044063097, −3.21633722467918311055067988645, −2.87540285562128618456317033555, −2.36189164010824430259825815892, −2.08200943282532521432217475762, −1.81407026564373330393774110499, −1.70519727751676029216613140611, −1.57771811047021672300334471524, −1.20558980700792052338484287360, −1.09038068875675641012416130284, −0.891014783860320532469911408234,
0.891014783860320532469911408234, 1.09038068875675641012416130284, 1.20558980700792052338484287360, 1.57771811047021672300334471524, 1.70519727751676029216613140611, 1.81407026564373330393774110499, 2.08200943282532521432217475762, 2.36189164010824430259825815892, 2.87540285562128618456317033555, 3.21633722467918311055067988645, 3.46308262692224247159044063097, 3.48217582897712719409779456911, 3.88901512924952851831933267317, 3.93851635189379161711132383221, 3.99660550496695633748179857107, 4.14127434968027192345883800648, 4.32872590968207422746806250725, 4.50789768954533815403045208375, 4.53865196170380346089488966297, 4.71227551592639260788695858267, 5.21488663269795335486138474278, 5.21562444954098450256128095981, 5.42800318493905304590272782572, 5.69417239012704625480313706885, 5.96718069782608620066582499746
