| L(s) = 1 | − 24·9-s + 32·17-s − 4·25-s + 20·49-s + 96·73-s + 324·81-s + 32·97-s − 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 768·153-s + 157-s + 163-s + 167-s + 56·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 8·9-s + 7.76·17-s − 4/5·25-s + 20/7·49-s + 11.2·73-s + 36·81-s + 3.24·97-s − 3.63·121-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 − 62.0·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.439027511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.439027511\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 3 | \( ( 1 + p T^{2} )^{8} \) |
| 11 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 4 T + p T^{2} )^{8} \) |
| 19 | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 42 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 66 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 126 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 12 T + p T^{2} )^{8} \) |
| 79 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 4 T + p T^{2} )^{8} \) |
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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
−3.49376772648709768518113918282, −3.43615670721728573484216482868, −3.43203251232413174003467445541, −3.43144337232193972570936386942, −3.32658280375089690592564447430, −3.23856992150028245952581665077, −3.08714535313144076249349072865, −3.04020738241542363434072417661, −2.90786703816612024534591529841, −2.63145318787618838619906193945, −2.61734267802295081312562122963, −2.38140966551746300664480787465, −2.24272246280163187360359375747, −2.22861624234354253888874488207, −2.13817853071674380442672057233, −1.99919280279129028870866810718, −1.97798408771781646243855333806, −1.23284156326546853118167496338, −1.22070792569754039858983283889, −1.03020638287261687169096720594, −0.919918536663050212440372084544, −0.835041563737367004751385581521, −0.57174592993702278031863806562, −0.48025296122531240236736813808, −0.20719532136542557881851946381,
0.20719532136542557881851946381, 0.48025296122531240236736813808, 0.57174592993702278031863806562, 0.835041563737367004751385581521, 0.919918536663050212440372084544, 1.03020638287261687169096720594, 1.22070792569754039858983283889, 1.23284156326546853118167496338, 1.97798408771781646243855333806, 1.99919280279129028870866810718, 2.13817853071674380442672057233, 2.22861624234354253888874488207, 2.24272246280163187360359375747, 2.38140966551746300664480787465, 2.61734267802295081312562122963, 2.63145318787618838619906193945, 2.90786703816612024534591529841, 3.04020738241542363434072417661, 3.08714535313144076249349072865, 3.23856992150028245952581665077, 3.32658280375089690592564447430, 3.43144337232193972570936386942, 3.43203251232413174003467445541, 3.43615670721728573484216482868, 3.49376772648709768518113918282
Plot not available for L-functions of degree greater than 10.
