| L(s) = 1 | + 9-s + 12·11-s + 12·19-s + 18·29-s − 30·41-s + 2·49-s − 48·59-s + 22·61-s + 36·71-s + 32·79-s + 24·89-s + 12·99-s − 6·101-s + 62·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 12·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 1/3·9-s + 3.61·11-s + 2.75·19-s + 3.34·29-s − 4.68·41-s + 2/7·49-s − 6.24·59-s + 2.81·61-s + 4.27·71-s + 3.60·79-s + 2.54·89-s + 1.20·99-s − 0.597·101-s + 5.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1/13·169-s + 0.917·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.41966307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.41966307\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 25 T^{2} + 336 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 73 T^{2} + p^{2} T^{4} )( 1 + 26 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2^2$ | \( ( 1 + 15 T + 116 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 82 T^{2} + 4875 T^{4} + 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 24 T + 251 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 26 T^{2} - 3813 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 18 T + 179 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 119 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 12 T + 137 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 - 146 T^{2} + 11907 T^{4} - 146 p^{2} T^{6} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.27921077320131059898028835756, −5.86675727211625305247882154316, −5.54375350487451544969342282507, −5.20408267506900605656612628555, −4.99196741793218107333842651322, −4.93318164578763267904140991715, −4.92796196288650346848943078598, −4.91463512843707839081591927158, −4.19527203759458373512553777429, −4.18392229850462173970366848583, −3.83088583308546530330151116368, −3.73811837227445951296346428135, −3.68403467798065213441666100141, −3.25551099213900940515652727611, −3.15268254320185385511326973366, −2.92208755414724993806852547651, −2.88623083809645933447566225496, −2.19779766624414773853651537212, −2.05337724654186311721836780861, −1.74798157773867004761182366356, −1.48476557222975568496454965536, −1.28660893170518008855490555786, −1.06789524101850561347062097451, −0.73394203814052007625153112480, −0.52285067999517523214366981128,
0.52285067999517523214366981128, 0.73394203814052007625153112480, 1.06789524101850561347062097451, 1.28660893170518008855490555786, 1.48476557222975568496454965536, 1.74798157773867004761182366356, 2.05337724654186311721836780861, 2.19779766624414773853651537212, 2.88623083809645933447566225496, 2.92208755414724993806852547651, 3.15268254320185385511326973366, 3.25551099213900940515652727611, 3.68403467798065213441666100141, 3.73811837227445951296346428135, 3.83088583308546530330151116368, 4.18392229850462173970366848583, 4.19527203759458373512553777429, 4.91463512843707839081591927158, 4.92796196288650346848943078598, 4.93318164578763267904140991715, 4.99196741793218107333842651322, 5.20408267506900605656612628555, 5.54375350487451544969342282507, 5.86675727211625305247882154316, 6.27921077320131059898028835756
