| L(s) = 1 | + 4-s + 2·9-s + 16-s − 2·19-s + 6·25-s + 2·36-s + 18·43-s + 6·47-s − 6·49-s − 12·59-s + 64-s − 8·67-s − 2·76-s − 5·81-s + 18·83-s + 24·89-s + 6·100-s − 6·101-s − 18·103-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 2/3·9-s + 1/4·16-s − 0.458·19-s + 6/5·25-s + 1/3·36-s + 2.74·43-s + 0.875·47-s − 6/7·49-s − 1.56·59-s + 1/8·64-s − 0.977·67-s − 0.229·76-s − 5/9·81-s + 1.97·83-s + 2.54·89-s + 3/5·100-s − 0.597·101-s − 1.77·103-s + 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/6·144-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195364 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.068464020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.068464020\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 + p T^{2} \) |
| 17 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 120 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 48 T^{2} + p^{2} T^{4} \) |
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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
−9.160025348108921351005727814663, −8.756942805022124848298409715946, −7.986813940716329283809796267027, −7.65587088668776506942585074128, −7.24811733169512396588291526889, −6.65966732532351866812391117022, −6.26370099721229970828023941604, −5.76301756388933689619530604886, −5.08180300610355499200586820841, −4.50013557354371053098837084146, −4.05890927806236970190699935434, −3.25730355379818314650391441499, −2.64626039628269555711061537078, −1.91085072921640392396861172153, −0.970818847478859526690702316139,
0.970818847478859526690702316139, 1.91085072921640392396861172153, 2.64626039628269555711061537078, 3.25730355379818314650391441499, 4.05890927806236970190699935434, 4.50013557354371053098837084146, 5.08180300610355499200586820841, 5.76301756388933689619530604886, 6.26370099721229970828023941604, 6.65966732532351866812391117022, 7.24811733169512396588291526889, 7.65587088668776506942585074128, 7.986813940716329283809796267027, 8.756942805022124848298409715946, 9.160025348108921351005727814663
