| L(s) = 1 | − 2·16-s − 64·67-s − 144·101-s − 128·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 88·169-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 + ⋯ |
| L(s) = 1 | − 1/2·16-s − 7.81·67-s − 14.3·101-s − 12.6·103-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 + 6.76·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4470311869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4470311869\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{4} )^{2} \) |
| 17 | \( 1 \) |
| good | 3 | \( ( 1 - 8 T^{2} + 32 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )( 1 + 8 T^{2} + 32 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 5 | \( ( 1 + p^{4} T^{8} )^{2} \) |
| 7 | \( 1 + 4034 T^{8} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 24 T^{2} + 288 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )( 1 + 24 T^{2} + 288 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 13 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 238 T^{4} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 - 1809406 T^{8} + p^{8} T^{16} \) |
| 37 | \( 1 - 3356446 T^{8} + p^{8} T^{16} \) |
| 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 | \( ( 1 - p T^{2} )^{8} \) |
| 53 | \( ( 1 - 718 T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 61 | \( 1 - 13297246 T^{8} + p^{8} T^{16} \) |
| 67 | \( ( 1 + 8 T + p T^{2} )^{8} \) |
| 71 | \( ( 1 + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 48 T^{2} + 1152 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )( 1 + 48 T^{2} + 1152 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 79 | \( 1 - 64606846 T^{8} + p^{8} T^{16} \) |
| 83 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( 1 + 176908034 T^{8} + p^{8} T^{16} \) |
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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
−4.63880412886332289795380643043, −4.34578096171308629720812868309, −4.26999237902438684428827481811, −4.25396918588014835502066789728, −4.13761324265205908385737050625, −4.01377637796734567490896583539, −3.91963719539283711279995875594, −3.88331564778140735433892285895, −3.85769980144400858831983875865, −3.05430478848587715500267968355, −2.99912227779041885373800417268, −2.99907826588688147534435151785, −2.88239727090700862930137875159, −2.82079590106760747036820582231, −2.78744541420805860262014084301, −2.77787254591535575543733657877, −2.09804999257561438455093869979, −2.09596934791002315746910879496, −1.76978784547308270676853613027, −1.42147892705082335695064735520, −1.40668436355057884906428254735, −1.39141994190132169982894463692, −1.29072450440226118455906353889, −0.40538379771668143387815218752, −0.15244819790162822843759249496,
0.15244819790162822843759249496, 0.40538379771668143387815218752, 1.29072450440226118455906353889, 1.39141994190132169982894463692, 1.40668436355057884906428254735, 1.42147892705082335695064735520, 1.76978784547308270676853613027, 2.09596934791002315746910879496, 2.09804999257561438455093869979, 2.77787254591535575543733657877, 2.78744541420805860262014084301, 2.82079590106760747036820582231, 2.88239727090700862930137875159, 2.99907826588688147534435151785, 2.99912227779041885373800417268, 3.05430478848587715500267968355, 3.85769980144400858831983875865, 3.88331564778140735433892285895, 3.91963719539283711279995875594, 4.01377637796734567490896583539, 4.13761324265205908385737050625, 4.25396918588014835502066789728, 4.26999237902438684428827481811, 4.34578096171308629720812868309, 4.63880412886332289795380643043
Plot not available for L-functions of degree greater than 10.
