| L(s) = 1 | − 10·9-s − 4·25-s + 20·29-s − 40·37-s + 43·81-s + 84·109-s + 56·113-s + 74·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 46·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 3.33·9-s − 4/5·25-s + 3.71·29-s − 6.57·37-s + 43/9·81-s + 8.04·109-s + 5.26·113-s + 6.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.109442381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.109442381\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + T^{2} )^{4} \) |
| 7 | \( 1 - 34 T^{4} + p^{4} T^{8} \) |
| good | 3 | \( ( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 37 T^{2} + 576 T^{4} - 37 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 23 T^{2} + 264 T^{4} - 23 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 3 p T^{2} + 1220 T^{4} - 3 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 16 T^{2} - 66 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 5 T + 56 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 48 T^{2} + 1310 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 80 T^{2} + 3774 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + T^{2} + 3420 T^{4} + p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 128 T^{2} + 8238 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 90 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 208 T^{2} + 20286 T^{4} - 208 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 131 T^{2} + 16104 T^{4} + 131 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 20 T^{2} - 3066 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 155 T^{2} + 12276 T^{4} - 155 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.68590847338802755381699131465, −3.46056025118851033205502102348, −3.44373831069238384260059347159, −3.39723564419530370096073152584, −3.19808507581188066338169505267, −3.12097510560604680922931731132, −3.11030484287245875526584759119, −3.09637975642206951693667615436, −2.93821001093719055505543612946, −2.72439388245439149759730316225, −2.58496047327609689175909685116, −2.31263021664597465325711340608, −2.14119608149430904801010165539, −2.12617256126189113206418810041, −2.04080899835291544764364851649, −1.83047091297745045650366766625, −1.77738403205017294913463867114, −1.64717286988964490764543520302, −1.55687443235877722866599115349, −1.07324979304431865436647775616, −0.831847029922086119251576576787, −0.68152847842491732306821513207, −0.53852635116341489117438947713, −0.46566066717529896754426956364, −0.28394487742746485842902385082,
0.28394487742746485842902385082, 0.46566066717529896754426956364, 0.53852635116341489117438947713, 0.68152847842491732306821513207, 0.831847029922086119251576576787, 1.07324979304431865436647775616, 1.55687443235877722866599115349, 1.64717286988964490764543520302, 1.77738403205017294913463867114, 1.83047091297745045650366766625, 2.04080899835291544764364851649, 2.12617256126189113206418810041, 2.14119608149430904801010165539, 2.31263021664597465325711340608, 2.58496047327609689175909685116, 2.72439388245439149759730316225, 2.93821001093719055505543612946, 3.09637975642206951693667615436, 3.11030484287245875526584759119, 3.12097510560604680922931731132, 3.19808507581188066338169505267, 3.39723564419530370096073152584, 3.44373831069238384260059347159, 3.46056025118851033205502102348, 3.68590847338802755381699131465
Plot not available for L-functions of degree greater than 10.
