L(s) = 1 | + 16-s − 4·25-s + 8·67-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 + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
L(s) = 1 | + 16-s − 4·25-s + 8·67-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 + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.08820794348\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08820794348\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T^{4} + T^{8} \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 11 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 13 | \( ( 1 + T^{2} )^{8} \) |
| 17 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 19 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 23 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 29 | \( ( 1 + T^{4} )^{4} \) |
| 31 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 37 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 41 | \( ( 1 + T^{2} )^{8} \) |
| 43 | \( ( 1 + T^{2} )^{8} \) |
| 47 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 53 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 59 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 61 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 67 | \( ( 1 - T + T^{2} )^{8} \) |
| 71 | \( ( 1 + T^{4} )^{4} \) |
| 73 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 79 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 83 | \( ( 1 + T^{2} )^{8} \) |
| 89 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 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
−3.73620790976095700264468687548, −3.66324775930324827858933426828, −3.64685249252020382249505325622, −3.46883528625518137505445595394, −3.37037091654776793124497741623, −3.26899926929455470983045365633, −3.05552205755259613898249554663, −2.98589715333756599241069348626, −2.79066412955536811545513521573, −2.58966208542478317743268166827, −2.47615318281204067143685695725, −2.38614415516310752368940977267, −2.37038297846789174328206373223, −2.22464877370249704116077050575, −2.12752316644489962314444327526, −2.01885182563312885816428248709, −1.72795365529004449695429138533, −1.72432674491784480502389804280, −1.61020440235233608194342065241, −1.23844819419050847025256862902, −1.23423802939329208079921224439, −0.908463878619858256183076698821, −0.904979083194529587460774075689, −0.823239394549130815403278841678, −0.06837937282000647650238937318,
0.06837937282000647650238937318, 0.823239394549130815403278841678, 0.904979083194529587460774075689, 0.908463878619858256183076698821, 1.23423802939329208079921224439, 1.23844819419050847025256862902, 1.61020440235233608194342065241, 1.72432674491784480502389804280, 1.72795365529004449695429138533, 2.01885182563312885816428248709, 2.12752316644489962314444327526, 2.22464877370249704116077050575, 2.37038297846789174328206373223, 2.38614415516310752368940977267, 2.47615318281204067143685695725, 2.58966208542478317743268166827, 2.79066412955536811545513521573, 2.98589715333756599241069348626, 3.05552205755259613898249554663, 3.26899926929455470983045365633, 3.37037091654776793124497741623, 3.46883528625518137505445595394, 3.64685249252020382249505325622, 3.66324775930324827858933426828, 3.73620790976095700264468687548
Plot not available for L-functions of degree greater than 10.