| L(s) = 1 | + 32·67-s − 96·101-s + 64·103-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 + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
| L(s) = 1 | + 3.90·67-s − 9.55·101-s + 6.30·103-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 + 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.580997761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.580997761\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 158 T^{8} + p^{8} T^{16} \) |
| 5 | \( 1 + 866 T^{8} + p^{8} T^{16} \) |
| 7 | \( 1 + 1922 T^{8} + p^{8} T^{16} \) |
| 11 | \( 1 - 4318 T^{8} + p^{8} T^{16} \) |
| 13 | \( ( 1 - 6 T + p T^{2} )^{4}( 1 + 6 T + p T^{2} )^{4} \) |
| 19 | \( ( 1 - 238 T^{4} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 + 211202 T^{8} + p^{8} T^{16} \) |
| 29 | \( 1 - 745438 T^{8} + p^{8} T^{16} \) |
| 31 | \( 1 - 1846846 T^{8} + p^{8} T^{16} \) |
| 37 | \( 1 + 3726434 T^{8} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 96 T^{2} + 4608 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} )( 1 + 96 T^{2} + 4608 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 43 | \( ( 1 - 3214 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 718 T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 61 | \( 1 - 3268318 T^{8} + p^{8} T^{16} \) |
| 67 | \( ( 1 - 4 T + p T^{2} )^{8} \) |
| 71 | \( 1 - 48205438 T^{8} + p^{8} T^{16} \) |
| 73 | \( ( 1 + p^{4} T^{8} )^{2} \) |
| 79 | \( 1 - 27234238 T^{8} + p^{8} T^{16} \) |
| 83 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + p^{4} T^{8} )^{2} \) |
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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
−4.09919054121339133501738002511, −3.98681076109982089123113087245, −3.88282292298820470397090008608, −3.76883338914989497204753608832, −3.76051613080233454574408509613, −3.67972516873301222060377695210, −3.40619852971221464529889318577, −3.23088194204136420584313392897, −3.22050728981117804978518615738, −2.75438912550882716017122753861, −2.75201364187302737927268584695, −2.69514045218801122547243425705, −2.64113302177078909790804645019, −2.56582530339276467973152949660, −2.30064071676438148144528931831, −2.02933438406935455914836466031, −1.91929515258607134612459651130, −1.64776251618038806668236992763, −1.63747876573221260335087891957, −1.36251216319889987321742741897, −1.35543017657047645231078580036, −0.929764812442338811950167374737, −0.70690825037612477919879489731, −0.44070501417869422425077995296, −0.35054788757732288965735694127,
0.35054788757732288965735694127, 0.44070501417869422425077995296, 0.70690825037612477919879489731, 0.929764812442338811950167374737, 1.35543017657047645231078580036, 1.36251216319889987321742741897, 1.63747876573221260335087891957, 1.64776251618038806668236992763, 1.91929515258607134612459651130, 2.02933438406935455914836466031, 2.30064071676438148144528931831, 2.56582530339276467973152949660, 2.64113302177078909790804645019, 2.69514045218801122547243425705, 2.75201364187302737927268584695, 2.75438912550882716017122753861, 3.22050728981117804978518615738, 3.23088194204136420584313392897, 3.40619852971221464529889318577, 3.67972516873301222060377695210, 3.76051613080233454574408509613, 3.76883338914989497204753608832, 3.88282292298820470397090008608, 3.98681076109982089123113087245, 4.09919054121339133501738002511
Plot not available for L-functions of degree greater than 10.
