L(s) = 1 | − 2·4-s + 2·9-s + 16-s + 25-s − 4·36-s − 4·49-s + 81-s − 2·100-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 8·196-s + 197-s + ⋯ |
L(s) = 1 | − 2·4-s + 2·9-s + 16-s + 25-s − 4·36-s − 4·49-s + 81-s − 2·100-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 8·196-s + 197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2659127280\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2659127280\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 7 | \( ( 1 + T^{2} )^{4} \) |
| 17 | \( ( 1 + T^{2} )^{4} \) |
good | 2 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 3 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 11 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 13 | \( ( 1 + T^{2} )^{8} \) |
| 19 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 23 | \( ( 1 + T^{2} )^{8} \) |
| 29 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 31 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 37 | \( ( 1 + T^{2} )^{8} \) |
| 41 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 43 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 47 | \( ( 1 + T^{2} )^{8} \) |
| 53 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 59 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 61 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 67 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 71 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 73 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 79 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 83 | \( ( 1 + T^{2} )^{8} \) |
| 89 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 97 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
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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
−4.88525627028872056246986318221, −4.72549066111621973516151632524, −4.70488887014552577821441399093, −4.59917460442326889549399134182, −4.41344168131367716501653997964, −4.20880031637745118566691709463, −4.19559719899659002439716997438, −4.04179783920250744786937520009, −3.88479226252338813219282300852, −3.74503393369097399676145947786, −3.60604730450667242401418910635, −3.29685507612423088611875851878, −3.18011851165176019413257849082, −3.11460186069803654669947098457, −3.08777722823703857913177828321, −2.80079153919391625472958966902, −2.65436942874941726731857832039, −2.31779773827848594370283607151, −1.92323250903807836762478498063, −1.88100127907639550668405239851, −1.85398317211527482765458946232, −1.79079941191697797368039226207, −1.26116887579317186780080196876, −0.972231739966072897882836686047, −0.910485245674178209221107723569,
0.910485245674178209221107723569, 0.972231739966072897882836686047, 1.26116887579317186780080196876, 1.79079941191697797368039226207, 1.85398317211527482765458946232, 1.88100127907639550668405239851, 1.92323250903807836762478498063, 2.31779773827848594370283607151, 2.65436942874941726731857832039, 2.80079153919391625472958966902, 3.08777722823703857913177828321, 3.11460186069803654669947098457, 3.18011851165176019413257849082, 3.29685507612423088611875851878, 3.60604730450667242401418910635, 3.74503393369097399676145947786, 3.88479226252338813219282300852, 4.04179783920250744786937520009, 4.19559719899659002439716997438, 4.20880031637745118566691709463, 4.41344168131367716501653997964, 4.59917460442326889549399134182, 4.70488887014552577821441399093, 4.72549066111621973516151632524, 4.88525627028872056246986318221
Plot not available for L-functions of degree greater than 10.