L(s) = 1 | + 16-s + 4·23-s − 8·43-s + 4·53-s − 4·67-s − 4·107-s − 4·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
L(s) = 1 | + 16-s + 4·23-s − 8·43-s + 4·53-s − 4·67-s − 4·107-s − 4·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8101694338\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8101694338\) |
\(L(1)\) |
|
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 | \( 1 \) |
good | 3 | \( 1 - T^{8} + T^{16} \) |
| 5 | \( 1 - T^{8} + T^{16} \) |
| 11 | \( ( 1 - T^{2} + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 13 | \( ( 1 + T^{8} )^{2} \) |
| 17 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 19 | \( 1 - T^{8} + T^{16} \) |
| 23 | \( ( 1 - T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | \( ( 1 + T^{2} )^{4}( 1 + T^{4} )^{2} \) |
| 31 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 37 | \( ( 1 - T^{2} + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 41 | \( ( 1 + T^{4} )^{4} \) |
| 43 | \( ( 1 + T )^{8}( 1 + T^{4} )^{2} \) |
| 47 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 53 | \( ( 1 - T + T^{2} )^{4}( 1 - T^{4} + T^{8} ) \) |
| 59 | \( 1 - T^{8} + T^{16} \) |
| 61 | \( 1 - T^{8} + T^{16} \) |
| 67 | \( ( 1 + T + T^{2} )^{4}( 1 - T^{4} + T^{8} ) \) |
| 71 | \( ( 1 + T^{4} )^{4} \) |
| 73 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 79 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 83 | \( ( 1 + T^{8} )^{2} \) |
| 89 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 97 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
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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
−4.32091839715597638296715882756, −3.92480582293908730913424133634, −3.87850318134142175550693680608, −3.75830773395126790286800628107, −3.65832458456491740958238174288, −3.65269610480469941550797630183, −3.57955079712986227932811022244, −3.25333407088804613811006807467, −3.22950628708172012774376965815, −3.06916107337106384349818976731, −2.98273937392826191919327881469, −2.84058506932663814556917294984, −2.69519748392960791656496228892, −2.56460838447072322396931691092, −2.53998353275788361659527029498, −2.22596918028077293939360721292, −2.19954351432691813571606637911, −1.70933040767705908283164863611, −1.70656024992243671519218408500, −1.57330090913512116335388911329, −1.43890372712622404381595271969, −1.20258403451416158438986234569, −1.19447181610287746628002711232, −0.981520388703506868513534067892, −0.38957572206392821768195466533,
0.38957572206392821768195466533, 0.981520388703506868513534067892, 1.19447181610287746628002711232, 1.20258403451416158438986234569, 1.43890372712622404381595271969, 1.57330090913512116335388911329, 1.70656024992243671519218408500, 1.70933040767705908283164863611, 2.19954351432691813571606637911, 2.22596918028077293939360721292, 2.53998353275788361659527029498, 2.56460838447072322396931691092, 2.69519748392960791656496228892, 2.84058506932663814556917294984, 2.98273937392826191919327881469, 3.06916107337106384349818976731, 3.22950628708172012774376965815, 3.25333407088804613811006807467, 3.57955079712986227932811022244, 3.65269610480469941550797630183, 3.65832458456491740958238174288, 3.75830773395126790286800628107, 3.87850318134142175550693680608, 3.92480582293908730913424133634, 4.32091839715597638296715882756
Plot not available for L-functions of degree greater than 10.