L(s) = 1 | − 2·3-s + 4-s + 7-s + 9-s − 2·12-s − 13-s + 16-s + 2·19-s − 2·21-s + 8·25-s + 28-s + 31-s + 36-s − 37-s + 2·39-s − 3·43-s − 2·48-s + 49-s − 52-s − 4·57-s + 2·61-s + 63-s − 67-s − 73-s − 16·75-s + 2·76-s − 6·79-s + ⋯ |
L(s) = 1 | − 2·3-s + 4-s + 7-s + 9-s − 2·12-s − 13-s + 16-s + 2·19-s − 2·21-s + 8·25-s + 28-s + 31-s + 36-s − 37-s + 2·39-s − 3·43-s − 2·48-s + 49-s − 52-s − 4·57-s + 2·61-s + 63-s − 67-s − 73-s − 16·75-s + 2·76-s − 6·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3506908275\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3506908275\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 7 | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 31 | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
good | 2 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \) |
| 5 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 11 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \) |
| 13 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \) |
| 17 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 19 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \) |
| 23 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \) |
| 29 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 37 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \) |
| 41 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \) |
| 43 | \( ( 1 + T + T^{2} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \) |
| 47 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \) |
| 53 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \) |
| 59 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 61 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \) |
| 67 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \) |
| 71 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \) |
| 73 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \) |
| 79 | \( ( 1 + T + T^{2} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 83 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \) |
| 89 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \) |
| 97 | \( ( 1 + T + T^{2} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \) |
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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
−4.97974061117353615715670040519, −4.69372929808314072580179380949, −4.65671705208458787936416197834, −4.52327184358880997504128002947, −4.43749728738938755366429068697, −4.41696509610318933301576814636, −4.11902977849050498640915666317, −3.92535502280937876494365473092, −3.67639932264403124947333234355, −3.65871847298837649141404566174, −3.34799048177176697362990521535, −3.15415999135854336821940660760, −2.98903040439681826328278002316, −2.90687012267960322145783687691, −2.89407308767475916061983033335, −2.76682612996131169016455850082, −2.61129235269368708880184306515, −2.48940496781993318147992781903, −2.26035470487862325042572892934, −1.69307356906174333171882108280, −1.58627161361881869979310977079, −1.36983498626936051887762811829, −1.36023976477269823422789915915, −1.07658701612858209410588550522, −0.992944956544925699952571803408,
0.992944956544925699952571803408, 1.07658701612858209410588550522, 1.36023976477269823422789915915, 1.36983498626936051887762811829, 1.58627161361881869979310977079, 1.69307356906174333171882108280, 2.26035470487862325042572892934, 2.48940496781993318147992781903, 2.61129235269368708880184306515, 2.76682612996131169016455850082, 2.89407308767475916061983033335, 2.90687012267960322145783687691, 2.98903040439681826328278002316, 3.15415999135854336821940660760, 3.34799048177176697362990521535, 3.65871847298837649141404566174, 3.67639932264403124947333234355, 3.92535502280937876494365473092, 4.11902977849050498640915666317, 4.41696509610318933301576814636, 4.43749728738938755366429068697, 4.52327184358880997504128002947, 4.65671705208458787936416197834, 4.69372929808314072580179380949, 4.97974061117353615715670040519
Plot not available for L-functions of degree greater than 10.