| L(s) = 1 | + 2-s − 2·4-s − 5-s − 3·8-s − 10-s + 5·11-s + 4·13-s + 16-s + 11·17-s − 3·19-s + 2·20-s + 5·22-s − 2·23-s + 2·25-s + 4·26-s − 9·31-s + 2·32-s + 11·34-s − 4·37-s − 3·38-s + 3·40-s − 41-s − 10·43-s − 10·44-s − 2·46-s + 14·47-s − 9·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 4-s − 0.447·5-s − 1.06·8-s − 0.316·10-s + 1.50·11-s + 1.10·13-s + 1/4·16-s + 2.66·17-s − 0.688·19-s + 0.447·20-s + 1.06·22-s − 0.417·23-s + 2/5·25-s + 0.784·26-s − 1.61·31-s + 0.353·32-s + 1.88·34-s − 0.657·37-s − 0.486·38-s + 0.474·40-s − 0.156·41-s − 1.52·43-s − 1.50·44-s − 0.294·46-s + 2.04·47-s − 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57289761 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57289761 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.604785306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.604785306\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 29 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 5 T + 27 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 11 T + 63 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 29 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 9 T + 51 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + T + 71 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 106 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + T + 87 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - T + 121 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 158 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 14 T + 150 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + T + 127 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 87 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 137 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 13 T + 135 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
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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
−7.88459943534720628157692914467, −7.83537674822379891727120988647, −7.44017077797015206223541996258, −6.84037561779015388678618355210, −6.72017767568210285323956542016, −6.22673600252947180946836472461, −5.78410466072182894272959827141, −5.53140973067693156846699179620, −5.35783377548697195677269349012, −4.88793211929119153623588473287, −4.26173422144343011549242531135, −4.13267674993182048909767688163, −3.70633238115869278558748447001, −3.68208429220805701304412300669, −3.00317137124081724324657105516, −2.90569629734524821377489212328, −1.78482537912958157657611169535, −1.43571576194055332557475879467, −1.17282855779647595164503915840, −0.29756994223083770618652804036,
0.29756994223083770618652804036, 1.17282855779647595164503915840, 1.43571576194055332557475879467, 1.78482537912958157657611169535, 2.90569629734524821377489212328, 3.00317137124081724324657105516, 3.68208429220805701304412300669, 3.70633238115869278558748447001, 4.13267674993182048909767688163, 4.26173422144343011549242531135, 4.88793211929119153623588473287, 5.35783377548697195677269349012, 5.53140973067693156846699179620, 5.78410466072182894272959827141, 6.22673600252947180946836472461, 6.72017767568210285323956542016, 6.84037561779015388678618355210, 7.44017077797015206223541996258, 7.83537674822379891727120988647, 7.88459943534720628157692914467
