| L(s) = 1 | + 8·29-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 − 256-s + 257-s + ⋯ |
| L(s) = 1 | + 8·29-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 − 256-s + 257-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7907826355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7907826355\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T^{8} \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 + T^{4} )^{2} \) |
| 17 | \( ( 1 + T^{4} )^{2} \) |
| good | 7 | \( 1 + T^{16} \) |
| 11 | \( 1 + T^{16} \) |
| 13 | \( ( 1 + T^{8} )^{2} \) |
| 19 | \( ( 1 + T^{8} )^{2} \) |
| 23 | \( 1 + T^{16} \) |
| 29 | \( ( 1 - T )^{8}( 1 + T^{8} ) \) |
| 31 | \( 1 + T^{16} \) |
| 37 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 41 | \( ( 1 + T^{2} )^{4}( 1 + T^{8} ) \) |
| 43 | \( ( 1 + T^{8} )^{2} \) |
| 47 | \( ( 1 + T^{2} )^{8} \) |
| 53 | \( ( 1 + T^{2} )^{4}( 1 + T^{4} )^{2} \) |
| 59 | \( ( 1 + T^{8} )^{2} \) |
| 61 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 67 | \( ( 1 + T^{4} )^{4} \) |
| 71 | \( 1 + T^{16} \) |
| 73 | \( ( 1 + T^{2} )^{4}( 1 + T^{8} ) \) |
| 79 | \( 1 + T^{16} \) |
| 83 | \( ( 1 + T^{8} )^{2} \) |
| 89 | \( ( 1 + T^{8} )^{2} \) |
| 97 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
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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
−3.77806015706611001215571054924, −3.66338839674060073818231654665, −3.65810419945040790019033474347, −3.61141603376173451025798766753, −3.45612176893617532869175816392, −3.23976374658118449490784223886, −3.16660108259136848679618110500, −2.77686981984627690125155692838, −2.73545202929002323991041826986, −2.68944808985449873673357574086, −2.59505638367695085807755513238, −2.54584207210152543398688093505, −2.50268941121301805004356078913, −2.49848006038512025724821608898, −2.23252583972942574814418446340, −2.09101873633622426480803115837, −1.95629512739906003969993069339, −1.35821976799673983601946141663, −1.29486769753621016543006191476, −1.27529209317240236274961697361, −1.26333125832913263923903695914, −1.22689891936958999770443607087, −1.10215010150962598245580645059, −0.809856076376405585377816446305, −0.21261857510938718501849771957,
0.21261857510938718501849771957, 0.809856076376405585377816446305, 1.10215010150962598245580645059, 1.22689891936958999770443607087, 1.26333125832913263923903695914, 1.27529209317240236274961697361, 1.29486769753621016543006191476, 1.35821976799673983601946141663, 1.95629512739906003969993069339, 2.09101873633622426480803115837, 2.23252583972942574814418446340, 2.49848006038512025724821608898, 2.50268941121301805004356078913, 2.54584207210152543398688093505, 2.59505638367695085807755513238, 2.68944808985449873673357574086, 2.73545202929002323991041826986, 2.77686981984627690125155692838, 3.16660108259136848679618110500, 3.23976374658118449490784223886, 3.45612176893617532869175816392, 3.61141603376173451025798766753, 3.65810419945040790019033474347, 3.66338839674060073818231654665, 3.77806015706611001215571054924
Plot not available for L-functions of degree greater than 10.
