| L(s) = 1 | − 3·4-s + 2·5-s + 4·16-s + 12·19-s − 6·20-s + 5·25-s + 28·29-s − 4·31-s − 20·41-s + 13·49-s − 20·59-s − 14·61-s − 9·64-s + 8·71-s − 36·76-s − 4·79-s + 8·80-s − 18·89-s + 24·95-s − 15·100-s + 18·101-s + 10·109-s − 84·116-s + 22·121-s + 12·124-s + 22·125-s + 127-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 0.894·5-s + 16-s + 2.75·19-s − 1.34·20-s + 25-s + 5.19·29-s − 0.718·31-s − 3.12·41-s + 13/7·49-s − 2.60·59-s − 1.79·61-s − 9/8·64-s + 0.949·71-s − 4.12·76-s − 0.450·79-s + 0.894·80-s − 1.90·89-s + 2.46·95-s − 3/2·100-s + 1.79·101-s + 0.957·109-s − 7.79·116-s + 2·121-s + 1.07·124-s + 1.96·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.756223633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.756223633\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2^3$ | \( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 10 T^{2} - 1269 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 122 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 16 T + 183 T^{2} - 16 p T^{3} + p^{2} T^{4} )( 1 + 16 T + 183 T^{2} + 16 p T^{3} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 45 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 9 T - 8 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.698609719146941821046194015157, −8.109634480815861904650392522973, −8.045031713860872722403289987147, −8.040019877867038931166147554489, −7.35654422661224559653832176368, −7.16740253502427371740506492359, −7.02764703234062705145594354259, −6.65379077633378833328611523862, −6.40879592693807092142269954334, −6.00679510082260063121055803455, −5.91030439690509573297726387943, −5.44474803353433179002083624620, −5.29537782727193847823106833692, −4.83917553324183279065437108278, −4.72330931477009304620142931760, −4.53389543310639782100255310021, −4.43510199009314627191488005456, −3.50495866601066223137159886336, −3.36792355653108365370266870112, −3.25469664533210150890835989908, −2.66324550789052380189966411942, −2.53247835130626565846260934077, −1.48958335002481655634818506321, −1.35675966927314170876715232013, −0.70190913065472863437554892235,
0.70190913065472863437554892235, 1.35675966927314170876715232013, 1.48958335002481655634818506321, 2.53247835130626565846260934077, 2.66324550789052380189966411942, 3.25469664533210150890835989908, 3.36792355653108365370266870112, 3.50495866601066223137159886336, 4.43510199009314627191488005456, 4.53389543310639782100255310021, 4.72330931477009304620142931760, 4.83917553324183279065437108278, 5.29537782727193847823106833692, 5.44474803353433179002083624620, 5.91030439690509573297726387943, 6.00679510082260063121055803455, 6.40879592693807092142269954334, 6.65379077633378833328611523862, 7.02764703234062705145594354259, 7.16740253502427371740506492359, 7.35654422661224559653832176368, 8.040019877867038931166147554489, 8.045031713860872722403289987147, 8.109634480815861904650392522973, 8.698609719146941821046194015157
