| L(s) = 1 | − 3-s + 4-s + 5-s − 12-s − 15-s + 20-s + 8·23-s − 2·31-s + 2·47-s − 49-s + 2·53-s − 60-s − 8·69-s + 8·92-s + 2·93-s − 2·113-s + 8·115-s − 2·124-s + 127-s + 131-s + 137-s + 139-s − 2·141-s + 147-s + 149-s + 151-s − 2·155-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s + 5-s − 12-s − 15-s + 20-s + 8·23-s − 2·31-s + 2·47-s − 49-s + 2·53-s − 60-s − 8·69-s + 8·92-s + 2·93-s − 2·113-s + 8·115-s − 2·124-s + 127-s + 131-s + 137-s + 139-s − 2·141-s + 147-s + 149-s + 151-s − 2·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.762994634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.762994634\) |
| \(L(1)\) |
|
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 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 5 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 7 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 13 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 17 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 23 | $C_1$ | \( ( 1 - T )^{8} \) |
| 29 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 31 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 37 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 41 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 53 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 59 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 61 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 71 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 73 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 79 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 83 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.75976743168549194622050018298, −6.70936083732029149022481297475, −6.46240036686568621249048227451, −6.20629981546640164414514760135, −5.90537202730050813324192511362, −5.52970067268537253988029322906, −5.45847000480893646433914717494, −5.40022710397297976812335410404, −5.35899038042081557373083506983, −4.92806663331679141563735998933, −4.77869775618248663827716512182, −4.52182196098430655038893368790, −4.38750063675609260562580982950, −3.82768895340007164308622845790, −3.64951055694796944916546225112, −3.26711293045039118533479787536, −3.17829273602596063121315119260, −2.90248422150130880180294765483, −2.68851887938753586690287345124, −2.38992379716021432069759903252, −2.14810978352900364288800599568, −1.84673757713900894070225206380, −1.16989773128452195689913651419, −1.10845664957994227174785695838, −0.975605875279988683946736662856,
0.975605875279988683946736662856, 1.10845664957994227174785695838, 1.16989773128452195689913651419, 1.84673757713900894070225206380, 2.14810978352900364288800599568, 2.38992379716021432069759903252, 2.68851887938753586690287345124, 2.90248422150130880180294765483, 3.17829273602596063121315119260, 3.26711293045039118533479787536, 3.64951055694796944916546225112, 3.82768895340007164308622845790, 4.38750063675609260562580982950, 4.52182196098430655038893368790, 4.77869775618248663827716512182, 4.92806663331679141563735998933, 5.35899038042081557373083506983, 5.40022710397297976812335410404, 5.45847000480893646433914717494, 5.52970067268537253988029322906, 5.90537202730050813324192511362, 6.20629981546640164414514760135, 6.46240036686568621249048227451, 6.70936083732029149022481297475, 6.75976743168549194622050018298
