| L(s) = 1 | + 4·9-s − 8·13-s + 8·25-s + 64·37-s − 4·49-s − 24·61-s − 64·73-s + 6·81-s + 16·97-s − 32·109-s − 32·117-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 4/3·9-s − 2.21·13-s + 8/5·25-s + 10.5·37-s − 4/7·49-s − 3.07·61-s − 7.49·73-s + 2/3·81-s + 1.62·97-s − 3.06·109-s − 2.95·117-s − 0.727·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.015475640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.015475640\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 4 T^{2} + 10 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | \( ( 1 + T^{2} )^{4} \) |
| good | 5 | \( ( 1 - 4 T^{2} + 42 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 4 T^{2} + 54 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 2 T + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 28 T^{2} + 582 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 68 T^{2} + 1866 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 28 T^{2} + 1062 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 76 T^{2} + 2934 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 28 T^{2} + 390 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 16 T + 126 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 76 T^{2} + 3078 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + p T^{2} )^{8} \) |
| 53 | \( ( 1 - 124 T^{2} + 7734 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 76 T^{2} + 8298 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 6 T + 128 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 16 T + p T^{2} )^{4}( 1 + 16 T + p T^{2} )^{4} \) |
| 71 | \( ( 1 + 196 T^{2} + 17958 T^{4} + 196 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 16 T + 198 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 172 T^{2} + 21066 T^{4} + 172 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 20 T^{2} + 15750 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−5.09583962217256339159267967597, −4.99224634689886563230060889828, −4.80487561728565065761467431716, −4.55907040978619357302915374813, −4.50084093112577562481140778635, −4.41615224709421167720096430448, −4.36071979960187832236372226593, −4.33448054185760728079070956153, −4.07906452628281230306348856295, −3.86172037936718677182151993526, −3.81520541167975223319159680453, −3.27042541336623839488462742754, −3.05691902553690391640102960250, −3.04520749856013737828975980917, −3.00856012859279898987969620565, −2.64504144420863156286251031827, −2.58836993737451655143473234505, −2.41627212659632353591051346387, −2.24700403637624500328610290326, −2.17234179294619989637522550980, −1.36182935562868555799922961392, −1.33268306698427048677751121155, −1.25528713615084735062295993055, −1.05091088301444855998878843524, −0.32952280190290071339497563229,
0.32952280190290071339497563229, 1.05091088301444855998878843524, 1.25528713615084735062295993055, 1.33268306698427048677751121155, 1.36182935562868555799922961392, 2.17234179294619989637522550980, 2.24700403637624500328610290326, 2.41627212659632353591051346387, 2.58836993737451655143473234505, 2.64504144420863156286251031827, 3.00856012859279898987969620565, 3.04520749856013737828975980917, 3.05691902553690391640102960250, 3.27042541336623839488462742754, 3.81520541167975223319159680453, 3.86172037936718677182151993526, 4.07906452628281230306348856295, 4.33448054185760728079070956153, 4.36071979960187832236372226593, 4.41615224709421167720096430448, 4.50084093112577562481140778635, 4.55907040978619357302915374813, 4.80487561728565065761467431716, 4.99224634689886563230060889828, 5.09583962217256339159267967597
Plot not available for L-functions of degree greater than 10.
