| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 3·8-s − 2·9-s + 7·11-s + 12-s + 16-s − 4·17-s + 2·18-s + 4·19-s − 7·22-s − 6·23-s − 3·24-s − 6·25-s − 2·27-s − 10·29-s + 10·31-s + 32-s + 7·33-s + 4·34-s − 2·36-s − 37-s − 4·38-s − 3·43-s + 7·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.06·8-s − 2/3·9-s + 2.11·11-s + 0.288·12-s + 1/4·16-s − 0.970·17-s + 0.471·18-s + 0.917·19-s − 1.49·22-s − 1.25·23-s − 0.612·24-s − 6/5·25-s − 0.384·27-s − 1.85·29-s + 1.79·31-s + 0.176·32-s + 1.21·33-s + 0.685·34-s − 1/3·36-s − 0.164·37-s − 0.648·38-s − 0.457·43-s + 1.05·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6429588996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6429588996\) |
| \(L(\frac{3}{2})\) |
|
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_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 1163 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 12 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T - 30 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 - 9 T + 78 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 15 T + 192 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
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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
−17.9205601225, −17.5429788143, −17.1214829253, −16.5799881599, −15.9140449419, −15.3935282103, −14.9270271991, −14.3451447528, −13.8243429683, −13.5013016747, −12.3741301577, −11.8447510304, −11.5294970012, −11.0698477119, −9.83017645868, −9.52838456408, −9.08153187799, −8.39305014413, −7.92399284159, −6.80978126345, −6.40697302729, −5.63078759715, −4.18665131809, −3.38619760586, −2.09105213336,
2.09105213336, 3.38619760586, 4.18665131809, 5.63078759715, 6.40697302729, 6.80978126345, 7.92399284159, 8.39305014413, 9.08153187799, 9.52838456408, 9.83017645868, 11.0698477119, 11.5294970012, 11.8447510304, 12.3741301577, 13.5013016747, 13.8243429683, 14.3451447528, 14.9270271991, 15.3935282103, 15.9140449419, 16.5799881599, 17.1214829253, 17.5429788143, 17.9205601225
