L(s) = 1 | + 16-s − 8·25-s + 8·31-s + 4·49-s − 8·73-s − 8·79-s + 4·97-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
L(s) = 1 | + 16-s − 8·25-s + 8·31-s + 4·49-s − 8·73-s − 8·79-s + 4·97-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7323252502\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7323252502\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T^{4} + T^{8} \) |
| 3 | \( 1 \) |
| 13 | \( ( 1 + T^{2} )^{4} \) |
good | 5 | \( ( 1 + T^{2} )^{8} \) |
| 7 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 11 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 17 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 19 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 29 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 31 | \( ( 1 - T + T^{2} )^{8} \) |
| 37 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 41 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 43 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | \( ( 1 + T^{4} )^{4} \) |
| 53 | \( ( 1 + T^{4} )^{4} \) |
| 59 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 61 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 73 | \( ( 1 + T + T^{2} )^{8} \) |
| 79 | \( ( 1 + T + T^{2} )^{8} \) |
| 83 | \( ( 1 + T^{4} )^{4} \) |
| 89 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 97 | \( ( 1 - T )^{8}( 1 + T + T^{2} )^{4} \) |
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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
−3.95831116604501037524674309170, −3.76332016515124291872483293898, −3.70777447678610635101824094149, −3.45334501527198821090906291092, −3.38405683044417881380104918368, −3.37672896446304369296287223520, −3.20678841280352177865620280326, −2.97034609599529079434561117133, −2.81222511047038156345090485796, −2.79266029835728106789630031298, −2.65259674955848953701348172529, −2.64008714678031672743661496664, −2.40154837050265628514443653671, −2.30554743640382846796144802426, −2.16819207018575213623791876507, −2.12751864959224761131217339036, −1.88072038830117549913498291307, −1.70783543270450642179230711713, −1.48055083329666816100694649044, −1.36359621402942253837810313397, −1.30970017334041693668125799601, −1.14164366727025013572954536858, −1.02026587231404982824013366085, −0.65001964578343855080224952487, −0.25830311344138809117543851115,
0.25830311344138809117543851115, 0.65001964578343855080224952487, 1.02026587231404982824013366085, 1.14164366727025013572954536858, 1.30970017334041693668125799601, 1.36359621402942253837810313397, 1.48055083329666816100694649044, 1.70783543270450642179230711713, 1.88072038830117549913498291307, 2.12751864959224761131217339036, 2.16819207018575213623791876507, 2.30554743640382846796144802426, 2.40154837050265628514443653671, 2.64008714678031672743661496664, 2.65259674955848953701348172529, 2.79266029835728106789630031298, 2.81222511047038156345090485796, 2.97034609599529079434561117133, 3.20678841280352177865620280326, 3.37672896446304369296287223520, 3.38405683044417881380104918368, 3.45334501527198821090906291092, 3.70777447678610635101824094149, 3.76332016515124291872483293898, 3.95831116604501037524674309170
Plot not available for L-functions of degree greater than 10.