| L(s) = 1 | − 8·67-s + 8·107-s + 8·109-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 | − 8·67-s + 8·107-s + 8·109-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.03113141303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03113141303\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T^{8} \) |
| good | 3 | \( 1 + T^{16} \) |
| 5 | \( 1 + T^{16} \) |
| 11 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 13 | \( 1 + T^{16} \) |
| 17 | \( ( 1 + T^{4} )^{4} \) |
| 19 | \( 1 + T^{16} \) |
| 23 | \( ( 1 + T^{2} )^{4}( 1 + T^{4} )^{2} \) |
| 29 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 31 | \( ( 1 + T^{2} )^{8} \) |
| 37 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 41 | \( ( 1 + T^{8} )^{2} \) |
| 43 | \( ( 1 + T^{2} )^{4}( 1 + T^{8} ) \) |
| 47 | \( ( 1 + T^{4} )^{4} \) |
| 53 | \( ( 1 + T^{2} )^{4}( 1 + T^{8} ) \) |
| 59 | \( 1 + T^{16} \) |
| 61 | \( 1 + T^{16} \) |
| 67 | \( ( 1 + T )^{8}( 1 + T^{8} ) \) |
| 71 | \( ( 1 + T^{8} )^{2} \) |
| 73 | \( ( 1 + T^{8} )^{2} \) |
| 79 | \( ( 1 + T^{8} )^{2} \) |
| 83 | \( 1 + T^{16} \) |
| 89 | \( ( 1 + T^{8} )^{2} \) |
| 97 | \( ( 1 + T^{2} )^{8} \) |
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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
−3.54053310899835714143094394250, −3.49157427010735364500772090721, −3.41732769081169518564636350731, −3.35107889504442007055407548247, −3.32162454629884592162181488922, −3.18441893669126692677826933961, −3.15710681451149678206240784836, −3.03413749477570899494997355894, −2.95107037858127961539664201830, −2.73017174661721366016961377905, −2.53531971714350250057002713997, −2.32763638031431587320181007649, −2.27122566921199880228700095367, −2.14015438198263714535510167101, −2.06052663825748978372921775620, −1.94467740070951014045742488902, −1.94318244852493235812751151405, −1.68889835546375257805968472714, −1.58594520824121668992304161100, −1.17908716964799183519340592230, −1.12512123102319584129811584422, −1.05763875084802447877328725654, −0.973386327739859013508760469825, −0.73488706415226677483120727677, −0.04012982673344177023518815896,
0.04012982673344177023518815896, 0.73488706415226677483120727677, 0.973386327739859013508760469825, 1.05763875084802447877328725654, 1.12512123102319584129811584422, 1.17908716964799183519340592230, 1.58594520824121668992304161100, 1.68889835546375257805968472714, 1.94318244852493235812751151405, 1.94467740070951014045742488902, 2.06052663825748978372921775620, 2.14015438198263714535510167101, 2.27122566921199880228700095367, 2.32763638031431587320181007649, 2.53531971714350250057002713997, 2.73017174661721366016961377905, 2.95107037858127961539664201830, 3.03413749477570899494997355894, 3.15710681451149678206240784836, 3.18441893669126692677826933961, 3.32162454629884592162181488922, 3.35107889504442007055407548247, 3.41732769081169518564636350731, 3.49157427010735364500772090721, 3.54053310899835714143094394250
Plot not available for L-functions of degree greater than 10.
