| L(s) = 1 | − 3·3-s + 4-s − 5-s − 3·7-s + 4·9-s + 2·11-s − 3·12-s + 2·13-s + 3·15-s + 16-s − 3·17-s + 2·19-s − 20-s + 9·21-s + 6·25-s − 6·27-s − 3·28-s − 3·29-s − 31-s − 6·33-s + 3·35-s + 4·36-s + 3·37-s − 6·39-s − 12·41-s + 2·43-s + 2·44-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1/2·4-s − 0.447·5-s − 1.13·7-s + 4/3·9-s + 0.603·11-s − 0.866·12-s + 0.554·13-s + 0.774·15-s + 1/4·16-s − 0.727·17-s + 0.458·19-s − 0.223·20-s + 1.96·21-s + 6/5·25-s − 1.15·27-s − 0.566·28-s − 0.557·29-s − 0.179·31-s − 1.04·33-s + 0.507·35-s + 2/3·36-s + 0.493·37-s − 0.960·39-s − 1.87·41-s + 0.304·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2917880340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2917880340\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 151 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 10 T + p T^{2} ) \) |
| good | 3 | $C_4$ | \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T - p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 - T + 33 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3 T - 74 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 13 T + 142 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 - T + 165 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−19.6608383733, −19.2121435679, −18.4888559663, −18.0146774185, −17.1952031595, −16.7942066395, −16.3261722893, −15.8467366417, −15.2482264837, −14.4572972987, −13.2659745133, −12.9531760448, −11.9900495680, −11.6646656726, −11.0683029981, −10.4353750238, −9.58130447651, −8.66728109669, −7.32847054674, −6.56371103064, −6.09913286948, −5.04029808906, −3.59708676923,
3.59708676923, 5.04029808906, 6.09913286948, 6.56371103064, 7.32847054674, 8.66728109669, 9.58130447651, 10.4353750238, 11.0683029981, 11.6646656726, 11.9900495680, 12.9531760448, 13.2659745133, 14.4572972987, 15.2482264837, 15.8467366417, 16.3261722893, 16.7942066395, 17.1952031595, 18.0146774185, 18.4888559663, 19.2121435679, 19.6608383733
