| L(s) = 1 | + 2·2-s + 2·3-s − 4-s + 2·5-s + 4·6-s + 2·7-s − 8·8-s + 2·9-s + 4·10-s − 2·12-s + 6·13-s + 4·14-s + 4·15-s − 7·16-s + 10·17-s + 4·18-s − 8·19-s − 2·20-s + 4·21-s − 16·24-s + 2·25-s + 12·26-s + 6·27-s − 2·28-s − 10·29-s + 8·30-s − 16·31-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s − 1/2·4-s + 0.894·5-s + 1.63·6-s + 0.755·7-s − 2.82·8-s + 2/3·9-s + 1.26·10-s − 0.577·12-s + 1.66·13-s + 1.06·14-s + 1.03·15-s − 7/4·16-s + 2.42·17-s + 0.942·18-s − 1.83·19-s − 0.447·20-s + 0.872·21-s − 3.26·24-s + 2/5·25-s + 2.35·26-s + 1.15·27-s − 0.377·28-s − 1.85·29-s + 1.46·30-s − 2.87·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.451895492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.451895492\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| 29 | $C_2$ | \( 1 + 10 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−11.80395893303249768870519676218, −11.18862800854985526695845039945, −10.48890575667998024340343144108, −10.43850355827165764727384679476, −9.380646477905724250087981803222, −9.248408778112047765282570151290, −8.975491550479458637530119574214, −8.495490708247363347274491759847, −7.83363667711391691560098629410, −7.68236970326708885687680771375, −6.59635706735278448453341171809, −5.93049840184771123362171264523, −5.57834661030834213496804668705, −5.48085480227904536532493385280, −4.45255203170281525085139515379, −4.05525051534589975199287996640, −3.60554211954618019285764577600, −3.10963940476883368419109494757, −2.25582783374174827420636874708, −1.28962639369078622387977177308,
1.28962639369078622387977177308, 2.25582783374174827420636874708, 3.10963940476883368419109494757, 3.60554211954618019285764577600, 4.05525051534589975199287996640, 4.45255203170281525085139515379, 5.48085480227904536532493385280, 5.57834661030834213496804668705, 5.93049840184771123362171264523, 6.59635706735278448453341171809, 7.68236970326708885687680771375, 7.83363667711391691560098629410, 8.495490708247363347274491759847, 8.975491550479458637530119574214, 9.248408778112047765282570151290, 9.380646477905724250087981803222, 10.43850355827165764727384679476, 10.48890575667998024340343144108, 11.18862800854985526695845039945, 11.80395893303249768870519676218
