L(s) = 1 | + 4·5-s + 16-s + 6·25-s + 4·37-s + 4·53-s + 4·73-s + 4·80-s − 8·97-s − 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + 181-s + 16·185-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
L(s) = 1 | + 4·5-s + 16-s + 6·25-s + 4·37-s + 4·53-s + 4·73-s + 4·80-s − 8·97-s − 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + 181-s + 16·185-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(7.538307730\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.538307730\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T^{4} + T^{8} \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T^{4} + T^{8} \) |
good | 3 | \( 1 - T^{8} + T^{16} \) |
| 5 | \( ( 1 - T + T^{2} )^{4}( 1 - T^{4} + T^{8} ) \) |
| 11 | \( 1 - T^{8} + T^{16} \) |
| 13 | \( ( 1 + T^{2} )^{8} \) |
| 19 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 23 | \( 1 - T^{8} + T^{16} \) |
| 29 | \( ( 1 + T^{2} )^{4}( 1 + T^{4} )^{2} \) |
| 31 | \( 1 - T^{8} + T^{16} \) |
| 37 | \( ( 1 - T + T^{2} )^{4}( 1 - T^{4} + T^{8} ) \) |
| 41 | \( ( 1 + T^{2} )^{4}( 1 + T^{4} )^{2} \) |
| 43 | \( ( 1 + T^{4} )^{4} \) |
| 47 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 53 | \( ( 1 - T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 61 | \( ( 1 - T^{2} + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 67 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 71 | \( ( 1 + T^{8} )^{2} \) |
| 73 | \( ( 1 - T + T^{2} )^{4}( 1 - T^{4} + T^{8} ) \) |
| 79 | \( 1 - T^{8} + T^{16} \) |
| 83 | \( ( 1 + T^{4} )^{4} \) |
| 89 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 97 | \( ( 1 + T )^{8}( 1 + T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.78637001134685705429494871417, −3.71169838320893329692353108403, −3.63309268528160370259010312089, −3.42124044840710510385749605313, −3.27575185498820870117200127696, −3.00331807577184086874212112397, −2.99095660707273757757103185042, −2.87600148813840099622962838192, −2.75081087970167207113178591314, −2.71977948903573184038203509576, −2.45105361586228812973313585210, −2.38651959579585647361453192512, −2.34005726015665350144263961367, −2.31023400830011018112165371143, −2.18579476032607385604497716962, −2.03042327918176741754861910213, −1.84296255894846897247438665781, −1.69873463642145819251897627896, −1.36983094788062613065812156125, −1.34992039060709542781108489212, −1.31903512894637559127898255580, −1.22432762265950719599851836811, −1.08661392276431142217986290002, −0.67888575579658642926529406486, −0.57743158349372447365651190460,
0.57743158349372447365651190460, 0.67888575579658642926529406486, 1.08661392276431142217986290002, 1.22432762265950719599851836811, 1.31903512894637559127898255580, 1.34992039060709542781108489212, 1.36983094788062613065812156125, 1.69873463642145819251897627896, 1.84296255894846897247438665781, 2.03042327918176741754861910213, 2.18579476032607385604497716962, 2.31023400830011018112165371143, 2.34005726015665350144263961367, 2.38651959579585647361453192512, 2.45105361586228812973313585210, 2.71977948903573184038203509576, 2.75081087970167207113178591314, 2.87600148813840099622962838192, 2.99095660707273757757103185042, 3.00331807577184086874212112397, 3.27575185498820870117200127696, 3.42124044840710510385749605313, 3.63309268528160370259010312089, 3.71169838320893329692353108403, 3.78637001134685705429494871417
Plot not available for L-functions of degree greater than 10.