| L(s) = 1 | − 4·2-s + 8·4-s − 8·8-s − 4·16-s − 20·29-s + 32·32-s + 12·37-s + 10·49-s + 16·53-s + 80·58-s − 64·64-s − 48·74-s − 18·81-s − 40·98-s − 64·106-s − 60·109-s − 44·113-s − 160·116-s − 6·121-s + 127-s + 64·128-s + 131-s + 137-s + 139-s + 96·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4·4-s − 2.82·8-s − 16-s − 3.71·29-s + 5.65·32-s + 1.97·37-s + 10/7·49-s + 2.19·53-s + 10.5·58-s − 8·64-s − 5.57·74-s − 2·81-s − 4.04·98-s − 6.21·106-s − 5.74·109-s − 4.13·113-s − 14.8·116-s − 0.545·121-s + 0.0887·127-s + 5.65·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 7.89·148-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08531349876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08531349876\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 3 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 45 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 125 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 117 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 140 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 77 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 172 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 140 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.53466264477991861895961651012, −7.37878809346440969740632087470, −7.31940703077611429305866168551, −6.97407892081086405061520456432, −6.82057437359769679309617442531, −6.54963878103986925820443929845, −6.31729025016174349733008810109, −5.83945417483856083674901699063, −5.75036633541710867329856760684, −5.41269047548155887650837347689, −5.25054175094122962337875330546, −5.07786503218918670005608996312, −4.34351713877144888435180303206, −4.20371664487996929524821622331, −4.07067729161332318580868348105, −3.96898635548993709838742504320, −3.42283279392700061331238687174, −2.92047365219331500960942939566, −2.54275857874761757624658307465, −2.46904096054098927735191071718, −1.96734277197133247207816654441, −1.72091051330494252790410592561, −1.19474953885959037910513850180, −0.996361160485018902780394461586, −0.15868532179061038181355790295,
0.15868532179061038181355790295, 0.996361160485018902780394461586, 1.19474953885959037910513850180, 1.72091051330494252790410592561, 1.96734277197133247207816654441, 2.46904096054098927735191071718, 2.54275857874761757624658307465, 2.92047365219331500960942939566, 3.42283279392700061331238687174, 3.96898635548993709838742504320, 4.07067729161332318580868348105, 4.20371664487996929524821622331, 4.34351713877144888435180303206, 5.07786503218918670005608996312, 5.25054175094122962337875330546, 5.41269047548155887650837347689, 5.75036633541710867329856760684, 5.83945417483856083674901699063, 6.31729025016174349733008810109, 6.54963878103986925820443929845, 6.82057437359769679309617442531, 6.97407892081086405061520456432, 7.31940703077611429305866168551, 7.37878809346440969740632087470, 7.53466264477991861895961651012
