L(s) = 1 | − 2·3-s − 4-s + 5-s − 3·7-s + 2·11-s + 2·12-s − 2·15-s + 16-s + 4·17-s − 5·19-s − 20-s + 6·21-s + 8·23-s + 25-s + 5·27-s + 3·28-s − 5·29-s − 6·31-s − 4·33-s − 3·35-s + 2·37-s − 3·41-s + 2·43-s − 2·44-s − 2·47-s − 2·48-s + 3·49-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 1/2·4-s + 0.447·5-s − 1.13·7-s + 0.603·11-s + 0.577·12-s − 0.516·15-s + 1/4·16-s + 0.970·17-s − 1.14·19-s − 0.223·20-s + 1.30·21-s + 1.66·23-s + 1/5·25-s + 0.962·27-s + 0.566·28-s − 0.928·29-s − 1.07·31-s − 0.696·33-s − 0.507·35-s + 0.328·37-s − 0.468·41-s + 0.304·43-s − 0.301·44-s − 0.291·47-s − 0.288·48-s + 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 708 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 708 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3253436263\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3253436263\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
good | 5 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - 2 T - 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + 7 T + 72 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 134 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 14 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−19.2635183129, −18.8983865543, −18.3672695104, −17.5272060415, −17.1046203043, −16.7104638256, −16.3624662849, −15.3501628981, −14.6579725608, −14.1645025024, −13.0779421625, −12.8868227016, −12.2001214421, −11.3216041667, −10.7981771025, −9.97320924666, −9.29156821780, −8.67647726032, −7.32177052208, −6.39986997081, −5.82876143655, −4.90977462270, −3.42333274451,
3.42333274451, 4.90977462270, 5.82876143655, 6.39986997081, 7.32177052208, 8.67647726032, 9.29156821780, 9.97320924666, 10.7981771025, 11.3216041667, 12.2001214421, 12.8868227016, 13.0779421625, 14.1645025024, 14.6579725608, 15.3501628981, 16.3624662849, 16.7104638256, 17.1046203043, 17.5272060415, 18.3672695104, 18.8983865543, 19.2635183129