| L(s) = 1 | + 3·2-s + 4·4-s + 3·8-s + 3·16-s + 10·25-s + 16·31-s + 6·32-s + 28·49-s + 30·50-s − 48·53-s + 48·62-s + 5·64-s + 32·79-s + 12·83-s + 84·98-s + 40·100-s − 144·106-s + 44·121-s + 64·124-s + 127-s − 9·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 2·4-s + 1.06·8-s + 3/4·16-s + 2·25-s + 2.87·31-s + 1.06·32-s + 4·49-s + 4.24·50-s − 6.59·53-s + 6.09·62-s + 5/8·64-s + 3.60·79-s + 1.31·83-s + 8.48·98-s + 4·100-s − 13.9·106-s + 4·121-s + 5.74·124-s + 0.0887·127-s − 0.795·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.95377107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.95377107\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} )( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 23 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 24 T + 245 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 61 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 6 T + 95 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
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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
−6.95100205140356444187744430170, −6.73523850207189735230801742730, −6.57534466888298976776563362080, −6.19162994925142198398895005348, −6.09435923790924374315070597104, −5.97437994177761548106012245067, −5.83416839749747996943143457941, −5.24356448898584255860761547124, −5.06062879459427450252550057721, −4.93402645766601344487891253028, −4.92902759301663603288812607948, −4.45804796536762651884301609342, −4.40526077983566490349286823705, −4.20332195217683557140825594548, −3.83816086020409974160535109657, −3.41792681776504028841585461739, −3.31597749638716279274383074567, −3.04423014839780175853984046626, −2.94343898100509645340205630543, −2.48023690451748656439393152654, −2.27513587390936551727145601767, −1.81674584800059340149828537614, −1.43001160875703988676169465548, −0.851053034188434050893176677395, −0.68576196567743510256906648988,
0.68576196567743510256906648988, 0.851053034188434050893176677395, 1.43001160875703988676169465548, 1.81674584800059340149828537614, 2.27513587390936551727145601767, 2.48023690451748656439393152654, 2.94343898100509645340205630543, 3.04423014839780175853984046626, 3.31597749638716279274383074567, 3.41792681776504028841585461739, 3.83816086020409974160535109657, 4.20332195217683557140825594548, 4.40526077983566490349286823705, 4.45804796536762651884301609342, 4.92902759301663603288812607948, 4.93402645766601344487891253028, 5.06062879459427450252550057721, 5.24356448898584255860761547124, 5.83416839749747996943143457941, 5.97437994177761548106012245067, 6.09435923790924374315070597104, 6.19162994925142198398895005348, 6.57534466888298976776563362080, 6.73523850207189735230801742730, 6.95100205140356444187744430170
