| L(s) = 1 | + 4·2-s + 2·4-s − 20·8-s − 45·16-s + 16·17-s + 10·25-s + 16·32-s + 64·34-s − 16·47-s + 16·49-s + 40·50-s + 204·64-s + 32·68-s − 64·94-s + 64·98-s + 20·100-s + 32·109-s + 8·113-s + 32·121-s + 127-s + 232·128-s + 131-s − 320·136-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 4-s − 7.07·8-s − 11.2·16-s + 3.88·17-s + 2·25-s + 2.82·32-s + 10.9·34-s − 2.33·47-s + 16/7·49-s + 5.65·50-s + 51/2·64-s + 3.88·68-s − 6.60·94-s + 6.46·98-s + 2·100-s + 3.06·109-s + 0.752·113-s + 2.90·121-s + 0.0887·127-s + 20.5·128-s + 0.0873·131-s − 27.4·136-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.612381626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.612381626\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 7 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 128 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 112 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 104 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 114 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.64918673811559000011076625923, −6.39471052815351542809712046837, −6.20754767400767328610262705384, −6.00707744963655881905924045992, −5.90624231480792443259970507442, −5.50624087033319413760325716493, −5.47987684033867703766579037404, −5.12823014715231442412639266840, −5.11707910588242127616640712362, −5.05687793163365211985637042835, −4.48152972384568838221789783299, −4.44078267425656019976500251615, −4.42548651205483397013669454313, −3.85057418259126126021606185950, −3.62226176796495753702927879088, −3.51964660934055031843085276358, −3.37329178517369371306949452314, −3.19441745082094440985987217398, −2.85223241121986316373537135817, −2.65360676898071456553986060455, −2.37127697179203166620965126091, −1.52475655987613430901878308414, −1.15011577265868574015172391761, −0.77093851137323430412640739033, −0.47378398675577339982005819116,
0.47378398675577339982005819116, 0.77093851137323430412640739033, 1.15011577265868574015172391761, 1.52475655987613430901878308414, 2.37127697179203166620965126091, 2.65360676898071456553986060455, 2.85223241121986316373537135817, 3.19441745082094440985987217398, 3.37329178517369371306949452314, 3.51964660934055031843085276358, 3.62226176796495753702927879088, 3.85057418259126126021606185950, 4.42548651205483397013669454313, 4.44078267425656019976500251615, 4.48152972384568838221789783299, 5.05687793163365211985637042835, 5.11707910588242127616640712362, 5.12823014715231442412639266840, 5.47987684033867703766579037404, 5.50624087033319413760325716493, 5.90624231480792443259970507442, 6.00707744963655881905924045992, 6.20754767400767328610262705384, 6.39471052815351542809712046837, 6.64918673811559000011076625923
