| L(s) = 1 | − 2-s − 2·3-s + 2·4-s + 6·5-s + 2·6-s + 2·7-s − 5·8-s + 3·9-s − 6·10-s − 2·11-s − 4·12-s + 2·13-s − 2·14-s − 12·15-s + 5·16-s + 6·17-s − 3·18-s − 4·19-s + 12·20-s − 4·21-s + 2·22-s − 2·23-s + 10·24-s + 17·25-s − 2·26-s − 4·27-s + 4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 4-s + 2.68·5-s + 0.816·6-s + 0.755·7-s − 1.76·8-s + 9-s − 1.89·10-s − 0.603·11-s − 1.15·12-s + 0.554·13-s − 0.534·14-s − 3.09·15-s + 5/4·16-s + 1.45·17-s − 0.707·18-s − 0.917·19-s + 2.68·20-s − 0.872·21-s + 0.426·22-s − 0.417·23-s + 2.04·24-s + 17/5·25-s − 0.392·26-s − 0.769·27-s + 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.256731666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.256731666\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 26 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 109 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 110 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 97 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 129 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 62 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 194 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 222 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−12.47963995589029819295526310648, −11.70276715553557933844493548601, −11.50530040163972206601299999110, −10.82866716274895845714783537674, −10.36938564096962006674759496066, −10.18835345295472751493961142304, −9.577806794640621249754897573339, −9.364021806975089841297799209259, −8.524475969918972472559858011629, −8.161939317897230989109662638355, −7.30743486816592193257801011398, −6.53380845063870952667786544858, −6.38517709956669484565179440345, −5.75369936884517779737088399372, −5.45725405618762075598023319108, −5.11749517832103761916153067282, −3.75271478128334122181597471475, −2.56982956210564980010893424414, −2.06016902653797192392454285505, −1.25745751196044836850926097415,
1.25745751196044836850926097415, 2.06016902653797192392454285505, 2.56982956210564980010893424414, 3.75271478128334122181597471475, 5.11749517832103761916153067282, 5.45725405618762075598023319108, 5.75369936884517779737088399372, 6.38517709956669484565179440345, 6.53380845063870952667786544858, 7.30743486816592193257801011398, 8.161939317897230989109662638355, 8.524475969918972472559858011629, 9.364021806975089841297799209259, 9.577806794640621249754897573339, 10.18835345295472751493961142304, 10.36938564096962006674759496066, 10.82866716274895845714783537674, 11.50530040163972206601299999110, 11.70276715553557933844493548601, 12.47963995589029819295526310648
