| L(s) = 1 | + 12·9-s + 10·16-s + 90·81-s + 127-s + 131-s + 137-s + 139-s + 120·144-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 + ⋯ |
| L(s) = 1 | + 4·9-s + 5/2·16-s + 10·81-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 10·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-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 + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.608173282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.608173282\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 - p T^{2} )^{4} \) |
| 13 | \( 1 \) |
| good | 2 | \( ( 1 - 5 T^{4} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 - 2 T^{4} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 + 190 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + p T^{2} )^{8} \) |
| 19 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + p T^{2} )^{8} \) |
| 29 | \( ( 1 - p T^{2} )^{8} \) |
| 31 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 2930 T^{4} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 4370 T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - p T^{2} )^{8} \) |
| 59 | \( ( 1 + 6910 T^{4} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 3790 T^{4} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 13730 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 9550 T^{4} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.87280803711042296969604880660, −4.67847535034410811164496398790, −4.33789755386919939016430319827, −4.32898921763266576370035679932, −4.29648928156349805873280485803, −4.10684717872662527560332285157, −3.98152206935158859395202247106, −3.77806944038316498668913512194, −3.68145883692429373569907853483, −3.43597584618396594139736561140, −3.37822004936429757339339350209, −3.37124666151352713988568052088, −3.21047575916616932078415665418, −2.76365181053601422618025901266, −2.48675580889204872709051792132, −2.44387846737510216682798653360, −2.38214065357645810577182725926, −2.19175530138706615587672789831, −1.77404227561851285183094316794, −1.47998144922730687466026414876, −1.43933373722273104205715592922, −1.30151455778190880268180611409, −1.22356552749257777189594829339, −0.895613408524568201116100160247, −0.41867206179546221929824922636,
0.41867206179546221929824922636, 0.895613408524568201116100160247, 1.22356552749257777189594829339, 1.30151455778190880268180611409, 1.43933373722273104205715592922, 1.47998144922730687466026414876, 1.77404227561851285183094316794, 2.19175530138706615587672789831, 2.38214065357645810577182725926, 2.44387846737510216682798653360, 2.48675580889204872709051792132, 2.76365181053601422618025901266, 3.21047575916616932078415665418, 3.37124666151352713988568052088, 3.37822004936429757339339350209, 3.43597584618396594139736561140, 3.68145883692429373569907853483, 3.77806944038316498668913512194, 3.98152206935158859395202247106, 4.10684717872662527560332285157, 4.29648928156349805873280485803, 4.32898921763266576370035679932, 4.33789755386919939016430319827, 4.67847535034410811164496398790, 4.87280803711042296969604880660
Plot not available for L-functions of degree greater than 10.
