L(s) = 1 | − 4-s − 4·5-s − 3·16-s − 12·17-s + 4·20-s + 6·25-s − 4·37-s − 4·41-s + 8·43-s + 8·47-s − 8·59-s + 7·64-s − 8·67-s + 12·68-s + 12·80-s + 48·85-s − 4·89-s − 6·100-s + 4·101-s + 28·109-s − 10·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 4·148-s + ⋯ |
L(s) = 1 | − 1/2·4-s − 1.78·5-s − 3/4·16-s − 2.91·17-s + 0.894·20-s + 6/5·25-s − 0.657·37-s − 0.624·41-s + 1.21·43-s + 1.16·47-s − 1.04·59-s + 7/8·64-s − 0.977·67-s + 1.45·68-s + 1.34·80-s + 5.20·85-s − 0.423·89-s − 3/5·100-s + 0.398·101-s + 2.68·109-s − 0.909·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.328·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3575884727\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3575884727\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.018312456434419527119744159153, −8.693390433543831577647523795630, −8.251469284589790909670122189063, −7.64380613206652059735072783819, −7.26089867011393708858561091827, −6.79678312336036188663270632781, −6.35393046739437426289631565092, −5.62583091530747883907100729659, −4.73546190050037463375370585314, −4.49878458502574972078691617427, −4.11723676863830659898666685882, −3.56199012513494909227187173112, −2.69641876559375287050960973286, −1.97440148384713200215076620060, −0.36811147113713012302502489118,
0.36811147113713012302502489118, 1.97440148384713200215076620060, 2.69641876559375287050960973286, 3.56199012513494909227187173112, 4.11723676863830659898666685882, 4.49878458502574972078691617427, 4.73546190050037463375370585314, 5.62583091530747883907100729659, 6.35393046739437426289631565092, 6.79678312336036188663270632781, 7.26089867011393708858561091827, 7.64380613206652059735072783819, 8.251469284589790909670122189063, 8.693390433543831577647523795630, 9.018312456434419527119744159153