| L(s) = 1 | − 3·4-s − 2·5-s + 6·16-s − 6·19-s + 6·20-s + 7·25-s − 16·29-s + 8·31-s − 4·41-s + 24·49-s + 4·59-s − 60·61-s − 10·64-s + 16·71-s + 18·76-s − 12·80-s − 28·89-s + 12·95-s − 21·100-s + 28·101-s + 48·109-s + 48·116-s − 46·121-s − 24·124-s + 4·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 0.894·5-s + 3/2·16-s − 1.37·19-s + 1.34·20-s + 7/5·25-s − 2.97·29-s + 1.43·31-s − 0.624·41-s + 24/7·49-s + 0.520·59-s − 7.68·61-s − 5/4·64-s + 1.89·71-s + 2.06·76-s − 1.34·80-s − 2.96·89-s + 1.23·95-s − 2.09·100-s + 2.78·101-s + 4.59·109-s + 4.45·116-s − 4.18·121-s − 2.15·124-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5824241101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5824241101\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2 T - 3 T^{2} - 24 T^{3} - 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | \( ( 1 + T )^{6} \) |
| good | 7 | \( 1 - 24 T^{2} + 248 T^{4} - 1802 T^{6} + 248 p^{2} T^{8} - 24 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + 23 T^{2} + 8 T^{3} + 23 p T^{4} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 40 T^{2} + 760 T^{4} - 10630 T^{6} + 760 p^{2} T^{8} - 40 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 84 T^{2} + 3176 T^{4} - 69242 T^{6} + 3176 p^{2} T^{8} - 84 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 40 T^{2} + 1320 T^{4} - 27850 T^{6} + 1320 p^{2} T^{8} - 40 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 + 8 T + 36 T^{2} + 54 T^{3} + 36 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 - 4 T + p T^{2} - 16 T^{3} + p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 106 T^{2} + 4519 T^{4} - 145228 T^{6} + 4519 p^{2} T^{8} - 106 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 2 T + 73 T^{2} + 264 T^{3} + 73 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 130 T^{2} + 9995 T^{4} - 515940 T^{6} + 9995 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 150 T^{2} + 11135 T^{4} - 587540 T^{6} + 11135 p^{2} T^{8} - 150 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 216 T^{2} + 23048 T^{4} - 28606 p T^{6} + 23048 p^{2} T^{8} - 216 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 2 T + 148 T^{2} - 156 T^{3} + 148 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 30 T + 473 T^{2} + 4552 T^{3} + 473 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 276 T^{2} + 37544 T^{4} - 3150398 T^{6} + 37544 p^{2} T^{8} - 276 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 8 T + 91 T^{2} - 120 T^{3} + 91 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 148 T^{2} + 1000 T^{4} + 680246 T^{6} + 1000 p^{2} T^{8} - 148 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 9 T^{2} + 880 T^{3} + 9 p T^{4} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 406 T^{2} + 73671 T^{4} - 7779028 T^{6} + 73671 p^{2} T^{8} - 406 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 14 T + 313 T^{2} + 2472 T^{3} + 313 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 282 T^{2} + 44495 T^{4} - 4912556 T^{6} + 44495 p^{2} T^{8} - 282 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.82738855732839334592875817909, −4.68960544519192410701815878075, −4.50589767078116898289654644293, −4.44409850700301929408574125473, −4.29972417363840354773689967441, −4.19622572265960806721647592331, −3.92988522936384379598296742714, −3.87341120059094212065213946626, −3.50952276958141562022999091371, −3.44596534494967404261942485544, −3.40827036480113921532900802360, −3.40740267045391422776335282650, −2.92008193208234860590462382480, −2.82121555046560636523554117726, −2.48524480456123389389524301589, −2.44976006038993097396497868064, −2.32088951410355913559779022034, −2.04597021495804723764983479900, −1.65405937994654737969683400125, −1.47114977567734108057537293399, −1.37550077997361184511887701718, −1.11747570046908030427770617744, −0.75945716094089720072358435861, −0.29177037184855514264854367761, −0.23878958562833541049766989489,
0.23878958562833541049766989489, 0.29177037184855514264854367761, 0.75945716094089720072358435861, 1.11747570046908030427770617744, 1.37550077997361184511887701718, 1.47114977567734108057537293399, 1.65405937994654737969683400125, 2.04597021495804723764983479900, 2.32088951410355913559779022034, 2.44976006038993097396497868064, 2.48524480456123389389524301589, 2.82121555046560636523554117726, 2.92008193208234860590462382480, 3.40740267045391422776335282650, 3.40827036480113921532900802360, 3.44596534494967404261942485544, 3.50952276958141562022999091371, 3.87341120059094212065213946626, 3.92988522936384379598296742714, 4.19622572265960806721647592331, 4.29972417363840354773689967441, 4.44409850700301929408574125473, 4.50589767078116898289654644293, 4.68960544519192410701815878075, 4.82738855732839334592875817909
Plot not available for L-functions of degree greater than 10.
