L(s) = 1 | + 2·4-s + 16-s − 8·49-s − 3·81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s − 16·196-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
L(s) = 1 | + 2·4-s + 16-s − 8·49-s − 3·81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s − 16·196-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9003459537\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9003459537\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 5 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
| 7 | \( ( 1 + T^{2} )^{8} \) |
good | 3 | \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \) |
| 11 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) |
| 13 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \) |
| 17 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \) |
| 19 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \) |
| 23 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) |
| 29 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) |
| 31 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) |
| 37 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \) |
| 41 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) |
| 43 | \( ( 1 + T^{2} )^{16} \) |
| 47 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \) |
| 53 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \) |
| 59 | \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \) |
| 61 | \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \) |
| 67 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \) |
| 71 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \) |
| 73 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \) |
| 79 | \( ( 1 + T^{2} )^{8}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 83 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \) |
| 89 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) |
| 97 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.67121815734096461547153839704, −2.63828586661358771614611818293, −2.62240078266992149702370769786, −2.58875751020626439030698305832, −2.53143505801144383627364182538, −2.30080141689865138661017673037, −2.17091458178505453995293788491, −2.13390926212324365417884007073, −2.08441212485587964019482867358, −1.99529831607470987842968213744, −1.89028893538651765078316343077, −1.82697230194712186374138068627, −1.79634821642104132526691535871, −1.72390749674610302089145511011, −1.68188351044133965966852190134, −1.58018785390690581375111556196, −1.45499583956826926060293213301, −1.45264672726051789130269908628, −1.25997963002199621386917701032, −1.23065065796331416808026768510, −1.13247035947316375386959047337, −0.928049194437025365814701901397, −0.848698346682820657225015781619, −0.60415230265030939009640867070, −0.36957256646041690495853063504,
0.36957256646041690495853063504, 0.60415230265030939009640867070, 0.848698346682820657225015781619, 0.928049194437025365814701901397, 1.13247035947316375386959047337, 1.23065065796331416808026768510, 1.25997963002199621386917701032, 1.45264672726051789130269908628, 1.45499583956826926060293213301, 1.58018785390690581375111556196, 1.68188351044133965966852190134, 1.72390749674610302089145511011, 1.79634821642104132526691535871, 1.82697230194712186374138068627, 1.89028893538651765078316343077, 1.99529831607470987842968213744, 2.08441212485587964019482867358, 2.13390926212324365417884007073, 2.17091458178505453995293788491, 2.30080141689865138661017673037, 2.53143505801144383627364182538, 2.58875751020626439030698305832, 2.62240078266992149702370769786, 2.63828586661358771614611818293, 2.67121815734096461547153839704
Plot not available for L-functions of degree greater than 10.