| L(s) = 1 | − 3·4-s + 2·5-s + 4·9-s + 6·16-s − 6·19-s − 6·20-s + 7·25-s + 16·29-s + 8·31-s − 12·36-s + 4·41-s + 8·45-s + 24·49-s − 4·59-s − 60·61-s − 10·64-s − 16·71-s + 18·76-s + 12·80-s − 8·81-s + 28·89-s − 12·95-s − 21·100-s − 28·101-s + 48·109-s − 48·116-s − 46·121-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 0.894·5-s + 4/3·9-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 − 2·36-s + 0.624·41-s + 1.19·45-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 − 8/9·81-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.595116923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.595116923\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 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 | 3 | \( 1 - 4 T^{2} + 8 p T^{4} - 62 T^{6} + 8 p^{3} T^{8} - 4 p^{4} T^{10} + p^{6} T^{12} \) |
| 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
−7.11904502160699953085000137102, −6.63990066743912451584119523402, −6.43802464655902387947760643199, −6.28759581605124835763336746226, −6.24733095303766015821591575478, −6.11052392352295261666510675627, −5.77390626129074940428910387873, −5.62401297475751619080272198727, −5.33354345485305962232251803850, −4.98418674247858018663533438450, −4.71097117098394054475879838198, −4.62643853548956378579857939981, −4.61589358589630215739542468215, −4.35754183437834371887024659292, −4.22455135236498366305667709228, −3.86729568264320067947768012528, −3.66373112377009733709320178132, −3.12557198292951037263457555753, −2.85465039987396600095188827587, −2.75719864027067956614449167884, −2.67283940965577670857071286722, −1.90255614503579768851563293398, −1.59486006614633122519785379543, −1.28029398731193236096716304515, −0.74776098080473919257872564369,
0.74776098080473919257872564369, 1.28029398731193236096716304515, 1.59486006614633122519785379543, 1.90255614503579768851563293398, 2.67283940965577670857071286722, 2.75719864027067956614449167884, 2.85465039987396600095188827587, 3.12557198292951037263457555753, 3.66373112377009733709320178132, 3.86729568264320067947768012528, 4.22455135236498366305667709228, 4.35754183437834371887024659292, 4.61589358589630215739542468215, 4.62643853548956378579857939981, 4.71097117098394054475879838198, 4.98418674247858018663533438450, 5.33354345485305962232251803850, 5.62401297475751619080272198727, 5.77390626129074940428910387873, 6.11052392352295261666510675627, 6.24733095303766015821591575478, 6.28759581605124835763336746226, 6.43802464655902387947760643199, 6.63990066743912451584119523402, 7.11904502160699953085000137102
Plot not available for L-functions of degree greater than 10.
