| L(s) = 1 | − 12·11-s + 16-s − 16·31-s − 48·41-s − 32·61-s + 9·81-s + 96·101-s + 34·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 12·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 3.61·11-s + 1/4·16-s − 2.87·31-s − 7.49·41-s − 4.09·61-s + 81-s + 9.55·101-s + 3.09·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s − 0.904·176-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1908565486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1908565486\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{4} + T^{8} \) |
| 3 | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 337 T^{4} + p^{4} T^{8} )( 1 + 191 T^{4} + p^{4} T^{8} ) \) |
| 17 | \( ( 1 + 47 T^{4} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( 1 + 958 T^{4} + 637923 T^{8} + 958 p^{4} T^{12} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 46 T^{2} + 1275 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 7 T + p T^{2} )^{4}( 1 + 11 T + p T^{2} )^{4} \) |
| 37 | \( ( 1 - 2062 T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 12 T + 89 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( 1 + 3577 T^{4} + 9376128 T^{8} + 3577 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( 1 + 1918 T^{4} - 1200957 T^{8} + 1918 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 4174 T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 43 T^{2} - 1632 T^{4} - 43 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 8809 T^{4} + p^{4} T^{8} )( 1 + 2903 T^{4} + p^{4} T^{8} ) \) |
| 71 | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 1054 T^{4} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2}( 1 + 131 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | \( 1 + 6553 T^{4} - 4516512 T^{8} + 6553 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 175 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( 1 + 16417 T^{4} + 180988608 T^{8} + 16417 p^{4} T^{12} + p^{8} T^{16} \) |
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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
−4.95091554055800327763926489964, −4.86284770496604200918936157666, −4.84996107510514202045537497253, −4.56482447860314277897722331012, −4.44111732952015781981362646042, −4.23302501151927788344949989748, −3.80459739735309075213788624186, −3.78313904990275324470123091931, −3.61448504303322913696759081148, −3.58412739069110919752260029644, −3.45065169928727962165776898290, −3.21181300232146192634877644100, −3.13986599142439314715877444946, −2.95737061952475509572288313280, −2.78546498501407582595729293404, −2.53896050480412485614984564258, −2.33231696849821724972722525335, −2.18159339764586540696604536865, −1.99624713374154052971030551586, −1.86935696868450519155710332313, −1.51062300258471178478728731320, −1.47228465709926853373782177300, −1.18577114716356761928953018236, −0.27706917331607085837990708046, −0.19224318159838953662073708944,
0.19224318159838953662073708944, 0.27706917331607085837990708046, 1.18577114716356761928953018236, 1.47228465709926853373782177300, 1.51062300258471178478728731320, 1.86935696868450519155710332313, 1.99624713374154052971030551586, 2.18159339764586540696604536865, 2.33231696849821724972722525335, 2.53896050480412485614984564258, 2.78546498501407582595729293404, 2.95737061952475509572288313280, 3.13986599142439314715877444946, 3.21181300232146192634877644100, 3.45065169928727962165776898290, 3.58412739069110919752260029644, 3.61448504303322913696759081148, 3.78313904990275324470123091931, 3.80459739735309075213788624186, 4.23302501151927788344949989748, 4.44111732952015781981362646042, 4.56482447860314277897722331012, 4.84996107510514202045537497253, 4.86284770496604200918936157666, 4.95091554055800327763926489964
Plot not available for L-functions of degree greater than 10.
