| L(s) = 1 | + 2·16-s − 8·53-s + 4·81-s − 16·113-s − 8·121-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 + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | + 2·16-s − 8·53-s + 4·81-s − 16·113-s − 8·121-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 + 229-s + 233-s + 239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{16} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{16} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1540889744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1540889744\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 5 | \( 1 - T^{8} + T^{16} \) |
| 7 | \( 1 \) |
| good | 3 | \( ( 1 - T^{4} + T^{8} )^{4} \) |
| 11 | \( ( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8} \) |
| 13 | \( ( 1 + T^{8} )^{4} \) |
| 17 | \( ( 1 - T^{8} + T^{16} )^{2} \) |
| 19 | \( ( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8} \) |
| 23 | \( ( 1 - T^{4} + T^{8} )^{4} \) |
| 29 | \( ( 1 + T^{4} )^{8} \) |
| 31 | \( ( 1 - T^{2} + T^{4} )^{8} \) |
| 37 | \( ( 1 - T^{4} + T^{8} )^{4} \) |
| 41 | \( ( 1 + T^{8} )^{4} \) |
| 43 | \( ( 1 + T^{4} )^{8} \) |
| 47 | \( ( 1 - T^{4} + T^{8} )^{4} \) |
| 53 | \( ( 1 + T + T^{2} )^{8}( 1 - T^{2} + T^{4} )^{4} \) |
| 59 | \( ( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8} \) |
| 61 | \( ( 1 - T^{8} + T^{16} )^{2} \) |
| 67 | \( ( 1 - T^{4} + T^{8} )^{4} \) |
| 71 | \( ( 1 - T )^{16}( 1 + T )^{16} \) |
| 73 | \( ( 1 - T^{8} + T^{16} )^{2} \) |
| 79 | \( ( 1 - T^{2} + T^{4} )^{8} \) |
| 83 | \( ( 1 + T^{4} )^{8} \) |
| 89 | \( ( 1 - T^{8} + T^{16} )^{2} \) |
| 97 | \( ( 1 + T^{8} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.90908789726601974994207346512, −2.87879930550531657527706266394, −2.81285987596997729614699326455, −2.67855138154227589989887985242, −2.57688574310364587128843940915, −2.47895911302942702865657782955, −2.38780350328171679006700652100, −2.32359592202046439146272450793, −2.31622087899353581938902121366, −2.26745984249529510201006063058, −2.25334923901345136494817088892, −1.91222798708511541510137391799, −1.90046720803874689887790378417, −1.80173767710441374026707575090, −1.70160829947482711061508608904, −1.52419860039426188853104628960, −1.43215556003379197628671877211, −1.37713985780495308752362656855, −1.35823294481175125934541260987, −1.26534666742680647653959682610, −1.25477056353931840363894589746, −1.22572677767808113077800538271, −0.966980744804699731186136184365, −0.886067886780049811381084181155, −0.25224155225769042694524968774,
0.25224155225769042694524968774, 0.886067886780049811381084181155, 0.966980744804699731186136184365, 1.22572677767808113077800538271, 1.25477056353931840363894589746, 1.26534666742680647653959682610, 1.35823294481175125934541260987, 1.37713985780495308752362656855, 1.43215556003379197628671877211, 1.52419860039426188853104628960, 1.70160829947482711061508608904, 1.80173767710441374026707575090, 1.90046720803874689887790378417, 1.91222798708511541510137391799, 2.25334923901345136494817088892, 2.26745984249529510201006063058, 2.31622087899353581938902121366, 2.32359592202046439146272450793, 2.38780350328171679006700652100, 2.47895911302942702865657782955, 2.57688574310364587128843940915, 2.67855138154227589989887985242, 2.81285987596997729614699326455, 2.87879930550531657527706266394, 2.90908789726601974994207346512
Plot not available for L-functions of degree greater than 10.
