L(s) = 1 | − 2-s + 4-s − 3·7-s − 8-s − 2·9-s + 3·11-s + 3·14-s + 16-s + 2·18-s − 3·22-s + 4·25-s − 3·28-s − 3·29-s − 32-s − 2·36-s − 16·37-s + 6·43-s + 3·44-s + 2·49-s − 4·50-s − 16·53-s + 3·56-s + 3·58-s + 6·63-s + 64-s + 14·67-s − 17·71-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.13·7-s − 0.353·8-s − 2/3·9-s + 0.904·11-s + 0.801·14-s + 1/4·16-s + 0.471·18-s − 0.639·22-s + 4/5·25-s − 0.566·28-s − 0.557·29-s − 0.176·32-s − 1/3·36-s − 2.63·37-s + 0.914·43-s + 0.452·44-s + 2/7·49-s − 0.565·50-s − 2.19·53-s + 0.400·56-s + 0.393·58-s + 0.755·63-s + 1/8·64-s + 1.71·67-s − 2.01·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 905912 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 905912 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7686793954\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7686793954\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 2311 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 11 T + p T^{2} ) \) |
good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 83 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
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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
−8.345091396994305392494824465136, −7.70294586549036820231984629031, −7.27829301591914464364757131300, −6.86540208515584494691986012802, −6.42166952150802440156639288790, −6.18917584985310136754343317524, −5.56023289584323837502274117207, −5.13860017769227937819470815265, −4.45937277820706098174970367924, −3.79581636149028805348011630480, −3.22757278532271184296103865191, −3.05478618529910279347682137915, −2.12668155135531716679521654173, −1.51148920451791087593790981360, −0.46281255612505343437665199150,
0.46281255612505343437665199150, 1.51148920451791087593790981360, 2.12668155135531716679521654173, 3.05478618529910279347682137915, 3.22757278532271184296103865191, 3.79581636149028805348011630480, 4.45937277820706098174970367924, 5.13860017769227937819470815265, 5.56023289584323837502274117207, 6.18917584985310136754343317524, 6.42166952150802440156639288790, 6.86540208515584494691986012802, 7.27829301591914464364757131300, 7.70294586549036820231984629031, 8.345091396994305392494824465136