| L(s) = 1 | + 2·4-s + 16·13-s + 3·16-s − 4·25-s − 16·37-s + 32·52-s + 16·61-s + 12·64-s + 48·73-s + 48·97-s − 8·100-s − 16·109-s − 48·121-s + 127-s + 131-s + 137-s + 139-s − 32·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 48·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 4-s + 4.43·13-s + 3/4·16-s − 4/5·25-s − 2.63·37-s + 4.43·52-s + 2.04·61-s + 3/2·64-s + 5.61·73-s + 4.87·97-s − 4/5·100-s − 1.53·109-s − 4.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.63·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.400294490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.400294490\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T^{2} + T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 + T^{2} )^{4} \) |
| good | 7 | \( ( 1 - 30 T^{4} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 24 T^{2} + 354 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | \( ( 1 + 52 T^{2} + 1606 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 44 T^{2} + 1654 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 84 T^{2} + 3558 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 92 T^{2} + 5302 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 76 T^{2} + 3814 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 68 T^{2} + 4726 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 56 T^{2} + 5154 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 108 T^{2} + 9846 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 60 T^{2} + 2790 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 12 T + 174 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 84 T^{2} + 14118 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 220 T^{2} + 23830 T^{4} + 220 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 320 T^{2} + 41314 T^{4} - 320 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−5.82609118789874247891310929020, −5.79543572949825276710885226399, −5.62665409272539290889757372668, −5.16573032951059718213932810305, −5.11049119843668401458977973289, −5.10996689154766058152743004267, −5.09487052258034031005307469927, −4.87157939910257338634966895829, −4.42177608480659611933881617295, −4.15421217201776381067177923535, −3.94811917144873871088255721478, −3.85890993164528106041357574452, −3.68035551267735984724410978728, −3.62139934538957270480617976242, −3.60366871200687945978236714997, −3.29336171064823203356279132584, −3.18268434641946101065646405870, −2.73276586123131304730690730483, −2.46688580335935611625897960926, −2.17022206156719609799753510098, −2.16131970122140972135150112451, −1.84722318949126075872138500483, −1.38726266314132513264078684581, −1.14288406890099305048925118572, −0.977227666693878819353306141720,
0.977227666693878819353306141720, 1.14288406890099305048925118572, 1.38726266314132513264078684581, 1.84722318949126075872138500483, 2.16131970122140972135150112451, 2.17022206156719609799753510098, 2.46688580335935611625897960926, 2.73276586123131304730690730483, 3.18268434641946101065646405870, 3.29336171064823203356279132584, 3.60366871200687945978236714997, 3.62139934538957270480617976242, 3.68035551267735984724410978728, 3.85890993164528106041357574452, 3.94811917144873871088255721478, 4.15421217201776381067177923535, 4.42177608480659611933881617295, 4.87157939910257338634966895829, 5.09487052258034031005307469927, 5.10996689154766058152743004267, 5.11049119843668401458977973289, 5.16573032951059718213932810305, 5.62665409272539290889757372668, 5.79543572949825276710885226399, 5.82609118789874247891310929020
Plot not available for L-functions of degree greater than 10.
