| L(s) = 1 | + 4·4-s + 4·16-s + 48·25-s + 24·37-s − 544·43-s − 16·64-s + 416·67-s + 80·79-s + 192·100-s + 224·109-s − 468·121-s + 127-s + 131-s + 137-s + 139-s + 96·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 760·169-s − 2.17e3·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 4-s + 1/4·16-s + 1.91·25-s + 0.648·37-s − 12.6·43-s − 1/4·64-s + 6.20·67-s + 1.01·79-s + 1.91·100-s + 2.05·109-s − 3.86·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.648·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 4.49·169-s − 12.6·172-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.271054880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.271054880\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 - 24 T^{2} - 49 T^{4} - 24 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 + 234 T^{2} + 40115 T^{4} + 234 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 + 190 T^{2} + p^{4} T^{4} )^{4} \) |
| 17 | \( ( 1 - 88 T^{2} - 75777 T^{4} - 88 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 130 T^{2} - 113421 T^{4} - 130 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 - 742 T^{2} + 270723 T^{4} - 742 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 - 1440 T^{2} + p^{4} T^{4} )^{4} \) |
| 31 | \( ( 1 - 1330 T^{2} + 845379 T^{4} - 1330 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 6 T - 1333 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 - 2696 T^{2} + p^{4} T^{4} )^{4} \) |
| 43 | \( ( 1 + 68 T + p^{2} T^{2} )^{8} \) |
| 47 | \( ( 1 - 318 T^{2} - 4778557 T^{4} - 318 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 + 3936 T^{2} + 7601615 T^{4} + 3936 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 + 2226 T^{2} - 7162285 T^{4} + 2226 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 + 2030 T^{2} - 9724941 T^{4} + 2030 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 104 T + 6327 T^{2} - 104 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 71 | \( ( 1 - 5082 T^{2} + p^{4} T^{4} )^{4} \) |
| 73 | \( ( 1 - 6958 T^{2} + 20015523 T^{4} - 6958 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 20 T - 5841 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 83 | \( ( 1 - p T )^{8}( 1 + p T )^{8} \) |
| 89 | \( ( 1 + 6888 T^{2} - 15297697 T^{4} + 6888 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 6194 T^{2} + p^{4} T^{4} )^{4} \) |
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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
−3.91930859606590407764382358481, −3.85990459193185262313121855604, −3.75970735164291301272584118157, −3.73543190073226806435463886339, −3.67847504842378959872122961915, −3.39488858137471908068839661866, −3.34208739692758926976986093155, −3.21742163670067792422123583075, −3.06878141510135072579327248785, −2.89952191342869223358400595114, −2.72355703862033232306605000369, −2.56031139558836062275173848111, −2.47200357603658603120221075048, −2.27569026282975830212324332409, −2.00999573340480649906420268416, −1.99847618784656688534900891110, −1.75503083594924765713417260702, −1.61182540129062827467796848433, −1.51962495220678314069275476005, −1.15675685157525120225859944366, −1.09927522507250647586110911712, −1.08745903542785869878728096240, −0.52152547177415502502665984134, −0.21348892855032669779970362077, −0.14649717904135378005776394874,
0.14649717904135378005776394874, 0.21348892855032669779970362077, 0.52152547177415502502665984134, 1.08745903542785869878728096240, 1.09927522507250647586110911712, 1.15675685157525120225859944366, 1.51962495220678314069275476005, 1.61182540129062827467796848433, 1.75503083594924765713417260702, 1.99847618784656688534900891110, 2.00999573340480649906420268416, 2.27569026282975830212324332409, 2.47200357603658603120221075048, 2.56031139558836062275173848111, 2.72355703862033232306605000369, 2.89952191342869223358400595114, 3.06878141510135072579327248785, 3.21742163670067792422123583075, 3.34208739692758926976986093155, 3.39488858137471908068839661866, 3.67847504842378959872122961915, 3.73543190073226806435463886339, 3.75970735164291301272584118157, 3.85990459193185262313121855604, 3.91930859606590407764382358481
Plot not available for L-functions of degree greater than 10.
