L(s) = 1 | − 3·5-s + 9-s − 13-s + 4·17-s + 4·25-s + 13·29-s + 13·37-s − 2·41-s − 3·45-s + 11·49-s − 6·53-s − 61-s + 3·65-s + 81-s − 12·85-s + 3·89-s − 16·97-s + 101-s − 16·109-s + 6·113-s − 117-s + 4·121-s + 3·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
L(s) = 1 | − 1.34·5-s + 1/3·9-s − 0.277·13-s + 0.970·17-s + 4/5·25-s + 2.41·29-s + 2.13·37-s − 0.312·41-s − 0.447·45-s + 11/7·49-s − 0.824·53-s − 0.128·61-s + 0.372·65-s + 1/9·81-s − 1.30·85-s + 0.317·89-s − 1.62·97-s + 0.0995·101-s − 1.53·109-s + 0.564·113-s − 0.0924·117-s + 4/11·121-s + 0.268·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.583374277\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.583374277\) |
\(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 137 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 69 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 60 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
show more | | |
show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.026802014630520671653715408224, −7.87425940651087755371933156224, −7.34509195780880983806592442989, −6.93151058253605863191606391699, −6.48605081841693221194532843048, −5.97129068214749854859911765938, −5.44546533120119529000881903557, −4.82159520187157499614667480629, −4.41828049388328298747161929655, −4.09005754673361885725886372499, −3.45205137486449060724300338562, −2.90618149062613367102728869470, −2.45170243133299848654504061126, −1.31008367417830698796566025216, −0.66232951374400095322139015060,
0.66232951374400095322139015060, 1.31008367417830698796566025216, 2.45170243133299848654504061126, 2.90618149062613367102728869470, 3.45205137486449060724300338562, 4.09005754673361885725886372499, 4.41828049388328298747161929655, 4.82159520187157499614667480629, 5.44546533120119529000881903557, 5.97129068214749854859911765938, 6.48605081841693221194532843048, 6.93151058253605863191606391699, 7.34509195780880983806592442989, 7.87425940651087755371933156224, 8.026802014630520671653715408224