| L(s) = 1 | − 2-s − 4-s − 4·5-s + 5·7-s + 8-s − 2·9-s + 4·10-s + 11-s + 2·13-s − 5·14-s + 3·16-s + 2·17-s + 2·18-s + 13·19-s + 4·20-s − 22-s + 23-s + 6·25-s − 2·26-s − 3·27-s − 5·28-s + 3·29-s − 31-s − 3·32-s − 2·34-s − 20·35-s + 2·36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 1.78·5-s + 1.88·7-s + 0.353·8-s − 2/3·9-s + 1.26·10-s + 0.301·11-s + 0.554·13-s − 1.33·14-s + 3/4·16-s + 0.485·17-s + 0.471·18-s + 2.98·19-s + 0.894·20-s − 0.213·22-s + 0.208·23-s + 6/5·25-s − 0.392·26-s − 0.577·27-s − 0.944·28-s + 0.557·29-s − 0.179·31-s − 0.530·32-s − 0.342·34-s − 3.38·35-s + 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126786 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126786 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8054285821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8054285821\) |
| \(L(\frac{3}{2})\) |
|
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_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 3 T + p T^{2} ) \) |
| 113 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 9 T + p T^{2} ) \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 - 5 T + 17 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 - 3 T + T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 3 T - 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 9 T + 64 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T - 31 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 - 7 T + 97 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 18 T + 166 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_4$ | \( 1 + 6 T + 106 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 96 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 9 T + 94 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 5 T + 115 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 128 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−14.0503976818, −13.4953017469, −13.0078044079, −12.0595785331, −11.9436528428, −11.7195069593, −11.2746268949, −11.0057240697, −10.2883785662, −9.76308779051, −9.31074460442, −8.71671522251, −8.29950240343, −8.06082459375, −7.69895861312, −7.32921346660, −6.63191052144, −5.57156775472, −5.17327771477, −4.94907821277, −3.91151894147, −3.66542810762, −3.00321506061, −1.54226277064, −0.794034402984,
0.794034402984, 1.54226277064, 3.00321506061, 3.66542810762, 3.91151894147, 4.94907821277, 5.17327771477, 5.57156775472, 6.63191052144, 7.32921346660, 7.69895861312, 8.06082459375, 8.29950240343, 8.71671522251, 9.31074460442, 9.76308779051, 10.2883785662, 11.0057240697, 11.2746268949, 11.7195069593, 11.9436528428, 12.0595785331, 13.0078044079, 13.4953017469, 14.0503976818
