| L(s) = 1 | − 2-s + 4-s + 2·5-s − 7-s − 3·8-s + 9-s − 2·10-s − 13-s + 14-s + 16-s − 7·17-s − 18-s + 8·19-s + 2·20-s − 3·23-s + 2·25-s + 26-s − 28-s + 29-s + 32-s + 7·34-s − 2·35-s + 36-s − 8·38-s − 6·40-s − 5·41-s − 10·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.632·10-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 1.69·17-s − 0.235·18-s + 1.83·19-s + 0.447·20-s − 0.625·23-s + 2/5·25-s + 0.196·26-s − 0.188·28-s + 0.185·29-s + 0.176·32-s + 1.20·34-s − 0.338·35-s + 1/6·36-s − 1.29·38-s − 0.948·40-s − 0.780·41-s − 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2799 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2799 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5608265576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5608265576\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 311 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 24 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + T + 8 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + 5 T + 60 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 78 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 9 T + 134 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 104 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T - 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 114 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−18.1045517924, −17.9372685233, −17.5450561368, −16.8385301870, −16.1616681961, −15.8825667159, −15.2477738060, −14.7703230176, −13.8152113490, −13.5919159696, −12.9642158540, −12.1819461118, −11.6479872111, −11.1218890806, −10.1921463826, −9.78917150846, −9.30392183702, −8.71538387996, −7.92996711809, −6.86570211034, −6.60222526968, −5.67886436818, −4.78853330622, −3.32358126239, −2.15495897171,
2.15495897171, 3.32358126239, 4.78853330622, 5.67886436818, 6.60222526968, 6.86570211034, 7.92996711809, 8.71538387996, 9.30392183702, 9.78917150846, 10.1921463826, 11.1218890806, 11.6479872111, 12.1819461118, 12.9642158540, 13.5919159696, 13.8152113490, 14.7703230176, 15.2477738060, 15.8825667159, 16.1616681961, 16.8385301870, 17.5450561368, 17.9372685233, 18.1045517924
