| L(s) = 1 | + 2·3-s + 9-s − 4·19-s − 4·25-s − 2·27-s + 24·29-s + 16·31-s − 8·37-s − 24·47-s + 12·53-s − 8·57-s + 24·59-s − 8·75-s − 4·81-s + 48·87-s + 32·93-s − 32·103-s − 4·109-s − 16·111-s + 24·113-s − 16·121-s + 127-s + 131-s + 137-s + 139-s − 48·141-s + 149-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s − 0.917·19-s − 4/5·25-s − 0.384·27-s + 4.45·29-s + 2.87·31-s − 1.31·37-s − 3.50·47-s + 1.64·53-s − 1.05·57-s + 3.12·59-s − 0.923·75-s − 4/9·81-s + 5.14·87-s + 3.31·93-s − 3.15·103-s − 0.383·109-s − 1.51·111-s + 2.25·113-s − 1.45·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4.04·141-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.097341671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.097341671\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 + 4 T^{2} - 9 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 16 T^{2} + 135 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 28 T^{2} + 495 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 8 T^{2} - 465 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 - 70 T^{2} - 429 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^3$ | \( 1 + 62 T^{2} - 2397 T^{4} + 62 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2^3$ | \( 1 + 28 T^{2} - 7137 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{2} \) |
| 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
−6.54003281268467509015359001142, −6.13629041932152484475017100787, −6.05069780533584279307470182180, −5.84123497538938702547979711556, −5.58680340987879251229659355677, −5.19749676002646902385245885126, −4.96995876513802842666023947098, −4.94571184166375123241280904396, −4.67776462004007765434434209295, −4.53139977905019249359862998975, −4.21430116328806511180577838792, −3.94634767280332566531180311099, −3.77276181828754482905310660868, −3.67776826144763687796944812303, −3.10105826847871707999946141974, −3.05397835980079863949377736976, −2.79891924089608919773572298022, −2.58042292628804287151491408201, −2.55254286854752106108348586723, −2.01305091326423089361143897044, −1.89476967003019695923989206926, −1.40841232619491363115114189053, −1.18010167642330169905627542302, −0.69245663614210051488359043586, −0.46234514186710302445555291119,
0.46234514186710302445555291119, 0.69245663614210051488359043586, 1.18010167642330169905627542302, 1.40841232619491363115114189053, 1.89476967003019695923989206926, 2.01305091326423089361143897044, 2.55254286854752106108348586723, 2.58042292628804287151491408201, 2.79891924089608919773572298022, 3.05397835980079863949377736976, 3.10105826847871707999946141974, 3.67776826144763687796944812303, 3.77276181828754482905310660868, 3.94634767280332566531180311099, 4.21430116328806511180577838792, 4.53139977905019249359862998975, 4.67776462004007765434434209295, 4.94571184166375123241280904396, 4.96995876513802842666023947098, 5.19749676002646902385245885126, 5.58680340987879251229659355677, 5.84123497538938702547979711556, 6.05069780533584279307470182180, 6.13629041932152484475017100787, 6.54003281268467509015359001142
