L(s) = 1 | + 2·9-s − 4·11-s − 4·17-s + 16·19-s + 2·81-s + 4·89-s − 8·99-s + 8·107-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8·153-s + 157-s + 163-s + 167-s + 32·171-s + 173-s + 179-s + 181-s + 16·187-s + 191-s + 193-s + 197-s + ⋯ |
L(s) = 1 | + 2·9-s − 4·11-s − 4·17-s + 16·19-s + 2·81-s + 4·89-s − 8·99-s + 8·107-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8·153-s + 157-s + 163-s + 167-s + 32·171-s + 173-s + 179-s + 181-s + 16·187-s + 191-s + 193-s + 197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 41^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 41^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(8.066767647\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.066767647\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 41 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
good | 3 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \) |
| 5 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \) |
| 7 | \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \) |
| 11 | \( ( 1 + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) |
| 13 | \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \) |
| 17 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \) |
| 19 | \( ( 1 - T )^{16}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \) |
| 23 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) |
| 29 | \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \) |
| 31 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) |
| 37 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) |
| 43 | \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \) |
| 47 | \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \) |
| 53 | \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \) |
| 59 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \) |
| 61 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \) |
| 67 | \( ( 1 + T^{2} )^{8}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \) |
| 71 | \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \) |
| 73 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \) |
| 79 | \( ( 1 + T^{8} )^{4} \) |
| 83 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) |
| 89 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \) |
| 97 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.48824002064177471489708110439, −2.30404473099880439250847413450, −2.29869643635782235375271154065, −2.25496748131626789499865708180, −2.12812078451391936026980986601, −1.99033908806534598892910370671, −1.97555992783528939400242195505, −1.93833180960777503756691244518, −1.93427777478118049394413853545, −1.70809894549912456835430078729, −1.68896284814401397582757836961, −1.68039439243504973593144484830, −1.56513411731062356299048508483, −1.51208296255266945103021235851, −1.34875312185986664693987292299, −1.34216396281428064912484892966, −1.13272427363044811115347449746, −1.08334614076416637415692934825, −1.01247520895196650385612058042, −0.871545341548785867056798636661, −0.838425697314791906933377868514, −0.76506924620652288445823004539, −0.68645731562341171906524016317, −0.67722598940912859673041771093, −0.63638761071980130821720222556,
0.63638761071980130821720222556, 0.67722598940912859673041771093, 0.68645731562341171906524016317, 0.76506924620652288445823004539, 0.838425697314791906933377868514, 0.871545341548785867056798636661, 1.01247520895196650385612058042, 1.08334614076416637415692934825, 1.13272427363044811115347449746, 1.34216396281428064912484892966, 1.34875312185986664693987292299, 1.51208296255266945103021235851, 1.56513411731062356299048508483, 1.68039439243504973593144484830, 1.68896284814401397582757836961, 1.70809894549912456835430078729, 1.93427777478118049394413853545, 1.93833180960777503756691244518, 1.97555992783528939400242195505, 1.99033908806534598892910370671, 2.12812078451391936026980986601, 2.25496748131626789499865708180, 2.29869643635782235375271154065, 2.30404473099880439250847413450, 2.48824002064177471489708110439
Plot not available for L-functions of degree greater than 10.