| L(s) = 1 | + 4-s + 4·7-s + 4·11-s + 16-s − 2·25-s + 4·28-s − 16·29-s − 12·37-s − 16·43-s + 4·44-s + 9·49-s − 20·53-s + 64-s + 8·67-s − 24·71-s + 16·77-s + 8·79-s − 2·100-s − 16·107-s + 8·109-s + 4·112-s + 4·113-s − 16·116-s + 5·121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.51·7-s + 1.20·11-s + 1/4·16-s − 2/5·25-s + 0.755·28-s − 2.97·29-s − 1.97·37-s − 2.43·43-s + 0.603·44-s + 9/7·49-s − 2.74·53-s + 1/8·64-s + 0.977·67-s − 2.84·71-s + 1.82·77-s + 0.900·79-s − 1/5·100-s − 1.54·107-s + 0.766·109-s + 0.377·112-s + 0.376·113-s − 1.48·116-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.50137576168953687108768694563, −7.20998797023745053729444681431, −6.79853855086926084541674182690, −6.28007088881936232182159165344, −5.88973762689013006108448201245, −5.25964646621742773894548541242, −5.10529030531332236282474829425, −4.48205645016970734304911158565, −3.97123923054971891475622805047, −3.48314749553827494253882755808, −3.11620292712776161277981887353, −2.01165372244914077330477604676, −1.71668187590965789455829830277, −1.46696501363205523734355212811, 0,
1.46696501363205523734355212811, 1.71668187590965789455829830277, 2.01165372244914077330477604676, 3.11620292712776161277981887353, 3.48314749553827494253882755808, 3.97123923054971891475622805047, 4.48205645016970734304911158565, 5.10529030531332236282474829425, 5.25964646621742773894548541242, 5.88973762689013006108448201245, 6.28007088881936232182159165344, 6.79853855086926084541674182690, 7.20998797023745053729444681431, 7.50137576168953687108768694563
