L(s) = 1 | − 16·43-s − 8·83-s + 8·107-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 + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
L(s) = 1 | − 16·43-s − 8·83-s + 8·107-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 + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{32} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{32} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1018889976\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1018889976\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T^{8} + T^{16} \) |
| 3 | \( 1 - T^{8} + T^{16} \) |
| 17 | \( 1 - T^{8} + T^{16} \) |
good | 5 | \( 1 - T^{16} + T^{32} \) |
| 7 | \( 1 - T^{16} + T^{32} \) |
| 11 | \( ( 1 - T^{2} + T^{4} )^{4}( 1 + T^{8} )^{2} \) |
| 13 | \( ( 1 - T^{4} + T^{8} )^{4} \) |
| 19 | \( ( 1 - T^{8} + T^{16} )^{2} \) |
| 23 | \( 1 - T^{16} + T^{32} \) |
| 29 | \( 1 - T^{16} + T^{32} \) |
| 31 | \( 1 - T^{16} + T^{32} \) |
| 37 | \( ( 1 + T^{16} )^{2} \) |
| 41 | \( ( 1 + T^{4} )^{4}( 1 - T^{8} + T^{16} ) \) |
| 43 | \( ( 1 + T )^{16}( 1 - T^{4} + T^{8} )^{2} \) |
| 47 | \( ( 1 - T^{4} + T^{8} )^{4} \) |
| 53 | \( ( 1 + T^{8} )^{4} \) |
| 59 | \( ( 1 + T^{2} )^{8}( 1 - T^{4} + T^{8} )^{2} \) |
| 61 | \( 1 - T^{16} + T^{32} \) |
| 67 | \( ( 1 + T^{8} )^{2}( 1 - T^{8} + T^{16} ) \) |
| 71 | \( ( 1 + T^{16} )^{2} \) |
| 73 | \( ( 1 - T^{4} + T^{8} )^{2}( 1 - T^{8} + T^{16} ) \) |
| 79 | \( 1 - T^{16} + T^{32} \) |
| 83 | \( ( 1 + T + T^{2} )^{8}( 1 - T^{4} + T^{8} )^{2} \) |
| 89 | \( ( 1 + T^{8} )^{4} \) |
| 97 | \( ( 1 + T^{2} )^{8}( 1 - T^{8} + T^{16} ) \) |
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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
−2.89721978836371625739454126375, −2.69691609445899603045608824293, −2.53200576275795315873008682711, −2.53179617526972211247145990739, −2.40999627629611670130844738662, −2.37413522639109099777356112612, −2.26933004679280757940272604713, −2.17152424372232444348032018282, −2.13512939875867708074888504415, −2.09865433863225657375618903637, −1.96214863401798239993717702003, −1.79693685946077799637299706370, −1.73649221990895365951214668968, −1.69297616516209418127227313449, −1.68873102546366555531317735430, −1.64374155176984921668211168363, −1.60964664061964359731485009065, −1.45783408385176911054375619124, −1.36133624318732821079596927104, −1.15996574714772711073154016525, −1.14324621730396476211768511027, −1.04297865486919240876483855920, −0.836786292120324365534751213634, −0.58101781224752806854027485419, −0.18406933482493440676209624857,
0.18406933482493440676209624857, 0.58101781224752806854027485419, 0.836786292120324365534751213634, 1.04297865486919240876483855920, 1.14324621730396476211768511027, 1.15996574714772711073154016525, 1.36133624318732821079596927104, 1.45783408385176911054375619124, 1.60964664061964359731485009065, 1.64374155176984921668211168363, 1.68873102546366555531317735430, 1.69297616516209418127227313449, 1.73649221990895365951214668968, 1.79693685946077799637299706370, 1.96214863401798239993717702003, 2.09865433863225657375618903637, 2.13512939875867708074888504415, 2.17152424372232444348032018282, 2.26933004679280757940272604713, 2.37413522639109099777356112612, 2.40999627629611670130844738662, 2.53179617526972211247145990739, 2.53200576275795315873008682711, 2.69691609445899603045608824293, 2.89721978836371625739454126375
Plot not available for L-functions of degree greater than 10.