L(s) = 1 | − 16·29-s − 8·41-s − 8·53-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 + 241-s + 251-s + ⋯ |
L(s) = 1 | − 16·29-s − 8·41-s − 8·53-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 + 241-s + 251-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.007617587542\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.007617587542\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T^{8} + T^{16} \) |
| 13 | \( 1 - T^{8} + T^{16} \) |
| 17 | \( ( 1 + T^{4} )^{4} \) |
good | 3 | \( 1 - T^{16} + T^{32} \) |
| 5 | \( ( 1 - T^{4} + T^{8} )^{2}( 1 - T^{8} + T^{16} ) \) |
| 7 | \( 1 - T^{16} + T^{32} \) |
| 11 | \( 1 - T^{16} + T^{32} \) |
| 19 | \( ( 1 - T^{8} + T^{16} )^{2} \) |
| 23 | \( 1 - T^{16} + T^{32} \) |
| 29 | \( ( 1 + T )^{16}( 1 - T^{8} + T^{16} ) \) |
| 31 | \( ( 1 + T^{16} )^{2} \) |
| 37 | \( ( 1 - T^{4} + T^{8} )^{2}( 1 + T^{8} )^{2} \) |
| 41 | \( ( 1 + T + T^{2} )^{8}( 1 + T^{8} )^{2} \) |
| 43 | \( ( 1 - T^{8} + T^{16} )^{2} \) |
| 47 | \( ( 1 + T^{2} )^{16} \) |
| 53 | \( ( 1 + T + T^{2} )^{8}( 1 - T^{4} + T^{8} )^{2} \) |
| 59 | \( ( 1 - T^{8} + T^{16} )^{2} \) |
| 61 | \( ( 1 - T^{4} + T^{8} )^{2}( 1 + T^{8} )^{2} \) |
| 67 | \( ( 1 - T^{4} + T^{8} )^{4} \) |
| 71 | \( 1 - T^{16} + T^{32} \) |
| 73 | \( ( 1 - T^{2} + T^{4} )^{4}( 1 - T^{8} + T^{16} ) \) |
| 79 | \( ( 1 + T^{16} )^{2} \) |
| 83 | \( ( 1 + T^{8} )^{4} \) |
| 89 | \( ( 1 - T^{8} + T^{16} )^{2} \) |
| 97 | \( ( 1 - T^{4} + T^{8} )^{2}( 1 - T^{8} + T^{16} ) \) |
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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
−3.01956291256799867276118492283, −2.96231531849742828239946534471, −2.95346579974800542305961535408, −2.62753160565342453313922888627, −2.56673584696770297072759354317, −2.55896967328168504622226507528, −2.53754628572748525109853490389, −2.22995995869731263848145539955, −2.19585735324789723093341616059, −2.10585389390308150948044667934, −2.04856665946721987106615956922, −1.88728834406980217021795888065, −1.88515118792304746813798763042, −1.87249548544910795489776618942, −1.79588994319590548235528916936, −1.68364917829225103476831952547, −1.65366081868462531482747031075, −1.58152813495542466743333154357, −1.52676982998994225240417367426, −1.47228763585329291814892710882, −1.39394239761761281693929727569, −1.35115960006997351740310417792, −0.821521010873783412990451230843, −0.52721632688693702275397830090, −0.095323343507969894942606698695,
0.095323343507969894942606698695, 0.52721632688693702275397830090, 0.821521010873783412990451230843, 1.35115960006997351740310417792, 1.39394239761761281693929727569, 1.47228763585329291814892710882, 1.52676982998994225240417367426, 1.58152813495542466743333154357, 1.65366081868462531482747031075, 1.68364917829225103476831952547, 1.79588994319590548235528916936, 1.87249548544910795489776618942, 1.88515118792304746813798763042, 1.88728834406980217021795888065, 2.04856665946721987106615956922, 2.10585389390308150948044667934, 2.19585735324789723093341616059, 2.22995995869731263848145539955, 2.53754628572748525109853490389, 2.55896967328168504622226507528, 2.56673584696770297072759354317, 2.62753160565342453313922888627, 2.95346579974800542305961535408, 2.96231531849742828239946534471, 3.01956291256799867276118492283
Plot not available for L-functions of degree greater than 10.