| L(s) = 1 | − 2·4-s + 16-s − 24·19-s + 48·31-s − 4·49-s − 6·64-s + 48·76-s − 56·109-s − 4·121-s − 96·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 16·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 8·196-s + 197-s + ⋯ |
| L(s) = 1 | − 4-s + 1/4·16-s − 5.50·19-s + 8.62·31-s − 4/7·49-s − 3/4·64-s + 5.50·76-s − 5.36·109-s − 0.363·121-s − 8.62·124-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 − 1.23·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 4/7·196-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4011040635\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4011040635\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 + T^{2} )^{4} \) |
| good | 2 | \( ( 1 + T^{2} + T^{4} + p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 2 T^{2} + 127 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 8 T^{2} + 238 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 32 T^{2} + 718 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 50 T^{2} + 1567 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 42 T^{2} + 1079 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 6 T + p T^{2} )^{8} \) |
| 37 | \( ( 1 - 65 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 20 T^{2} + 1606 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 82 T^{2} + 3523 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 24 T^{2} + 3518 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 68 T^{2} + 4918 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 24 T^{2} + 6062 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + p T^{2} )^{8} \) |
| 67 | \( ( 1 - 178 T^{2} + 15043 T^{4} - 178 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 90 T^{2} + 2711 T^{4} + 90 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 232 T^{2} + 23998 T^{4} - 232 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 129 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 36 T^{2} - 2602 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 160 T^{2} + 12846 T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{4} \) |
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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
−4.12613832004756132883808304498, −3.91835069558195567804720340168, −3.83655817688150360490591674380, −3.71007567426738839434975481127, −3.59925176854182469044382852484, −3.27079173610476546058065067689, −3.06541704021114377646044966162, −2.98637868398488236927110696357, −2.92296980519793725190952875481, −2.86275273938841274912824601712, −2.69118405787036053489710805693, −2.59905835998659698932263968360, −2.50150594184270198484881210026, −2.24535447504166971945491152603, −2.17381039338997613803193814309, −1.99046446173372709221805947382, −1.78612176003770298932610318727, −1.59999018950253437840123874649, −1.45234386963204189100433988025, −1.44979103011287694259612654848, −0.832905035012804275204871930223, −0.800552624707541126172825028039, −0.76174286644347241860624873776, −0.49646304807036853608405895212, −0.07609055042376660948463073696,
0.07609055042376660948463073696, 0.49646304807036853608405895212, 0.76174286644347241860624873776, 0.800552624707541126172825028039, 0.832905035012804275204871930223, 1.44979103011287694259612654848, 1.45234386963204189100433988025, 1.59999018950253437840123874649, 1.78612176003770298932610318727, 1.99046446173372709221805947382, 2.17381039338997613803193814309, 2.24535447504166971945491152603, 2.50150594184270198484881210026, 2.59905835998659698932263968360, 2.69118405787036053489710805693, 2.86275273938841274912824601712, 2.92296980519793725190952875481, 2.98637868398488236927110696357, 3.06541704021114377646044966162, 3.27079173610476546058065067689, 3.59925176854182469044382852484, 3.71007567426738839434975481127, 3.83655817688150360490591674380, 3.91835069558195567804720340168, 4.12613832004756132883808304498
Plot not available for L-functions of degree greater than 10.
