| L(s) = 1 | + 4·4-s + 12·16-s − 8·25-s − 28·49-s + 40·61-s + 32·64-s − 32·100-s + 80·109-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 20·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 112·196-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 2·4-s + 3·16-s − 8/5·25-s − 4·49-s + 5.12·61-s + 4·64-s − 3.19·100-s + 7.66·109-s − 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 8·196-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.425804130\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.425804130\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.563863182414027096217474413831, −8.966634101360067330824304167122, −8.529260969187460784632796263764, −8.377999266228609552005626490576, −8.258552870840540547541355627299, −7.88193980731212319209335734512, −7.57715513493759902669064024127, −7.24031602159848404724320040389, −7.19730794675868632149030313265, −6.83304990029392558771782532601, −6.36662993213246602994347229996, −6.19975010832180757297630976810, −6.15840013348621533193404551450, −5.70589676241628282685253829872, −5.19489457608380743712815347517, −5.11329766835867295657521775591, −4.73416679443851427953696469932, −4.05305370462435972202077182232, −3.69268178623577564891669876375, −3.50704247204928898210595539408, −3.08934152749611766974712122917, −2.56008608997485602523465379538, −2.04458946909688645023042934820, −1.95928357828876882028945803413, −1.12188090167904216822415215612,
1.12188090167904216822415215612, 1.95928357828876882028945803413, 2.04458946909688645023042934820, 2.56008608997485602523465379538, 3.08934152749611766974712122917, 3.50704247204928898210595539408, 3.69268178623577564891669876375, 4.05305370462435972202077182232, 4.73416679443851427953696469932, 5.11329766835867295657521775591, 5.19489457608380743712815347517, 5.70589676241628282685253829872, 6.15840013348621533193404551450, 6.19975010832180757297630976810, 6.36662993213246602994347229996, 6.83304990029392558771782532601, 7.19730794675868632149030313265, 7.24031602159848404724320040389, 7.57715513493759902669064024127, 7.88193980731212319209335734512, 8.258552870840540547541355627299, 8.377999266228609552005626490576, 8.529260969187460784632796263764, 8.966634101360067330824304167122, 9.563863182414027096217474413831
