| L(s) = 1 | + 4·4-s − 4·9-s + 12·16-s − 16·36-s + 8·49-s + 32·61-s + 32·64-s + 7·81-s + 64·109-s − 44·121-s + 127-s + 131-s + 137-s + 139-s − 48·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 32·196-s + ⋯ |
| L(s) = 1 | + 2·4-s − 4/3·9-s + 3·16-s − 8/3·36-s + 8/7·49-s + 4.09·61-s + 4·64-s + 7/9·81-s + 6.13·109-s − 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 16/7·196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.109932181\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.109932181\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
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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
−8.430185419055942607208072514309, −8.323886409467531876953720789446, −8.099889564467530799192740993572, −7.62346796558641939159658080491, −7.54662643931072457699598369477, −7.25271579290364933410954135621, −6.94745777030392306371897200553, −6.74966095109870690982018776324, −6.55760537413229617345144475204, −6.13738944215972820559193889449, −5.92609784288548712531003643413, −5.70347239274477912542262981235, −5.56387658201116029539267847275, −5.11244926398844520313607032892, −4.90793919715533224343542719492, −4.48114490032672114408415440880, −3.83745552114054671214763759206, −3.72870891484257363571015925605, −3.36592430985495431315241637102, −3.09350347636820903833734666410, −2.46962120675756535733012006642, −2.34879477507242529031403756799, −2.20948265216510993730510528330, −1.40846540252742740862468046415, −0.826896808199285001952706185328,
0.826896808199285001952706185328, 1.40846540252742740862468046415, 2.20948265216510993730510528330, 2.34879477507242529031403756799, 2.46962120675756535733012006642, 3.09350347636820903833734666410, 3.36592430985495431315241637102, 3.72870891484257363571015925605, 3.83745552114054671214763759206, 4.48114490032672114408415440880, 4.90793919715533224343542719492, 5.11244926398844520313607032892, 5.56387658201116029539267847275, 5.70347239274477912542262981235, 5.92609784288548712531003643413, 6.13738944215972820559193889449, 6.55760537413229617345144475204, 6.74966095109870690982018776324, 6.94745777030392306371897200553, 7.25271579290364933410954135621, 7.54662643931072457699598369477, 7.62346796558641939159658080491, 8.099889564467530799192740993572, 8.323886409467531876953720789446, 8.430185419055942607208072514309
