| L(s) = 1 | − 32·37-s + 32·43-s + 32·79-s − 32·109-s + 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 64·169-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 + ⋯ |
| L(s) = 1 | − 5.26·37-s + 4.87·43-s + 3.60·79-s − 3.06·109-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4.92·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3300486965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3300486965\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 + 48 T^{4} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 20 T^{2} + 214 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 32 T^{2} + 592 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 16 T^{2} + 64 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 4 T^{2} + 694 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 92 T^{2} + 3670 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 44 T^{2} + 2374 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 8 T + 88 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 96 T^{2} + 4608 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 108 T^{2} + 7302 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 156 T^{2} + 13014 T^{4} + 156 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 176 T^{2} + 14128 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 260 T^{2} + 26854 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 240 T^{2} + 24480 T^{4} - 240 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 12 T^{2} + 13302 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 192 T^{2} + 23136 T^{4} + 192 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 336 T^{2} + 46464 T^{4} - 336 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.22179582374106985687948057846, −3.19069248639252601088189347410, −3.18939853358993259507778788679, −2.77269385001125963418677357324, −2.72064048450740367774580464262, −2.67260383093966421076349675398, −2.51824631695324832158276186123, −2.47445277991148694205461537381, −2.26776952378465629430712287428, −2.25977559065271480881644531148, −2.12488819413898098326702039720, −2.05120183344966994532669281005, −1.87856591472453537333759649943, −1.86637093752516451751141309969, −1.52751249273918298430174049297, −1.52579825895698779457666885764, −1.29191832746272161862674408278, −1.27567932158929819208969679113, −1.03592345230120291762210919898, −0.936199096923585792646386629966, −0.882724409831705366553607951473, −0.76935020799118005436249496265, −0.34343656218144123503072271666, −0.22121468390301595157909422831, −0.05743720043484823956650769982,
0.05743720043484823956650769982, 0.22121468390301595157909422831, 0.34343656218144123503072271666, 0.76935020799118005436249496265, 0.882724409831705366553607951473, 0.936199096923585792646386629966, 1.03592345230120291762210919898, 1.27567932158929819208969679113, 1.29191832746272161862674408278, 1.52579825895698779457666885764, 1.52751249273918298430174049297, 1.86637093752516451751141309969, 1.87856591472453537333759649943, 2.05120183344966994532669281005, 2.12488819413898098326702039720, 2.25977559065271480881644531148, 2.26776952378465629430712287428, 2.47445277991148694205461537381, 2.51824631695324832158276186123, 2.67260383093966421076349675398, 2.72064048450740367774580464262, 2.77269385001125963418677357324, 3.18939853358993259507778788679, 3.19069248639252601088189347410, 3.22179582374106985687948057846
Plot not available for L-functions of degree greater than 10.
