| L(s) = 1 | − 2-s − 3·3-s + 5-s + 3·6-s + 7-s + 8-s + 3·9-s − 10-s + 2·11-s − 14-s − 3·15-s − 16-s + 4·17-s − 3·18-s + 6·19-s − 3·21-s − 2·22-s − 3·23-s − 3·24-s + 18·29-s + 3·30-s + 4·31-s − 6·33-s − 4·34-s + 35-s + 4·37-s − 6·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 0.447·5-s + 1.22·6-s + 0.377·7-s + 0.353·8-s + 9-s − 0.316·10-s + 0.603·11-s − 0.267·14-s − 0.774·15-s − 1/4·16-s + 0.970·17-s − 0.707·18-s + 1.37·19-s − 0.654·21-s − 0.426·22-s − 0.625·23-s − 0.612·24-s + 3.34·29-s + 0.547·30-s + 0.718·31-s − 1.04·33-s − 0.685·34-s + 0.169·35-s + 0.657·37-s − 0.973·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3851333659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3851333659\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 9 T + 14 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + T - 88 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−15.11319185182815227985695647685, −14.22179163347331973395479286369, −14.00886530662378715924612896919, −13.43321048577556451783639481960, −12.45153216538451057585401446535, −12.01490400259450215735294584831, −11.59553754720365939075670369519, −11.32160809582026924945694011020, −10.31547246634823912026371521725, −10.01876422695680704575392874056, −9.689065129551996658447007666514, −8.456590398572149363390766234868, −8.273069138088281899487541374925, −7.23818136825181266884753083364, −6.39523792820637389222306776151, −6.11722583319942204086745839998, −4.96869539482832325637312592115, −4.91249477605442021492006284165, −3.25116222547113681292297457491, −1.24616917281697553831129700746,
1.24616917281697553831129700746, 3.25116222547113681292297457491, 4.91249477605442021492006284165, 4.96869539482832325637312592115, 6.11722583319942204086745839998, 6.39523792820637389222306776151, 7.23818136825181266884753083364, 8.273069138088281899487541374925, 8.456590398572149363390766234868, 9.689065129551996658447007666514, 10.01876422695680704575392874056, 10.31547246634823912026371521725, 11.32160809582026924945694011020, 11.59553754720365939075670369519, 12.01490400259450215735294584831, 12.45153216538451057585401446535, 13.43321048577556451783639481960, 14.00886530662378715924612896919, 14.22179163347331973395479286369, 15.11319185182815227985695647685
