L(s) = 1 | − 8·4-s + 36·16-s − 120·64-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 | − 8·4-s + 36·16-s − 120·64-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 257^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 257^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.005620386069\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.005620386069\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( ( 1 + T^{2} )^{8} \) |
| 257 | \( ( 1 + T )^{16} \) |
good | 3 | \( 1 + T^{32} \) |
| 5 | \( ( 1 + T^{8} )^{2}( 1 + T^{16} ) \) |
| 7 | \( 1 + T^{32} \) |
| 11 | \( ( 1 + T^{8} )^{4} \) |
| 13 | \( ( 1 + T^{4} )^{4}( 1 + T^{8} )^{2} \) |
| 17 | \( ( 1 + T^{16} )^{2} \) |
| 19 | \( 1 + T^{32} \) |
| 23 | \( ( 1 + T^{8} )^{4} \) |
| 29 | \( ( 1 + T^{16} )^{2} \) |
| 31 | \( ( 1 + T^{16} )^{2} \) |
| 37 | \( ( 1 + T^{4} )^{4}( 1 + T^{16} ) \) |
| 41 | \( ( 1 + T^{8} )^{2}( 1 + T^{16} ) \) |
| 43 | \( 1 + T^{32} \) |
| 47 | \( 1 + T^{32} \) |
| 53 | \( ( 1 + T^{8} )^{2}( 1 + T^{16} ) \) |
| 59 | \( ( 1 + T^{16} )^{2} \) |
| 61 | \( ( 1 + T^{16} )^{2} \) |
| 67 | \( ( 1 + T^{8} )^{4} \) |
| 71 | \( 1 + T^{32} \) |
| 73 | \( ( 1 + T^{16} )^{2} \) |
| 79 | \( ( 1 + T^{16} )^{2} \) |
| 83 | \( 1 + T^{32} \) |
| 89 | \( ( 1 + T^{4} )^{4}( 1 + T^{8} )^{2} \) |
| 97 | \( ( 1 + T^{4} )^{4}( 1 + T^{16} ) \) |
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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
−2.88216865134159214637786775699, −2.83317436680373887543639993497, −2.80729428691038351153446275896, −2.69829654176970545030798434124, −2.68462711100802008411557846431, −2.55716486024389098907590815810, −2.50085070408490939972414146035, −2.37532307034652720143064142655, −2.17639511347170166686621380374, −2.08731970455126486790200488525, −2.02202125426084213664154871796, −1.83226508500775960564427770243, −1.82562943313818704567200651219, −1.78372075867876704961122071100, −1.77820330494842889831256155229, −1.37828472835755954226645719106, −1.33718969264319444772434139319, −1.21694136403485106242364710213, −1.21394696471838532946309467986, −1.07657613422300720616541096577, −1.06743200876436915685386767151, −0.951166093145613542744074044393, −0.869211434005572108358086717545, −0.64902038155353741455681052179, −0.12597442389847513736200472585,
0.12597442389847513736200472585, 0.64902038155353741455681052179, 0.869211434005572108358086717545, 0.951166093145613542744074044393, 1.06743200876436915685386767151, 1.07657613422300720616541096577, 1.21394696471838532946309467986, 1.21694136403485106242364710213, 1.33718969264319444772434139319, 1.37828472835755954226645719106, 1.77820330494842889831256155229, 1.78372075867876704961122071100, 1.82562943313818704567200651219, 1.83226508500775960564427770243, 2.02202125426084213664154871796, 2.08731970455126486790200488525, 2.17639511347170166686621380374, 2.37532307034652720143064142655, 2.50085070408490939972414146035, 2.55716486024389098907590815810, 2.68462711100802008411557846431, 2.69829654176970545030798434124, 2.80729428691038351153446275896, 2.83317436680373887543639993497, 2.88216865134159214637786775699
Plot not available for L-functions of degree greater than 10.