| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s + 6·5-s − 4·6-s − 4·7-s − 4·8-s + 2·9-s − 12·10-s + 6·12-s − 11·13-s + 8·14-s + 12·15-s + 5·16-s − 6·17-s − 4·18-s − 10·19-s + 18·20-s − 8·21-s − 2·23-s − 8·24-s + 17·25-s + 22·26-s + 6·27-s − 12·28-s + 15·29-s − 24·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s + 2.68·5-s − 1.63·6-s − 1.51·7-s − 1.41·8-s + 2/3·9-s − 3.79·10-s + 1.73·12-s − 3.05·13-s + 2.13·14-s + 3.09·15-s + 5/4·16-s − 1.45·17-s − 0.942·18-s − 2.29·19-s + 4.02·20-s − 1.74·21-s − 0.417·23-s − 1.63·24-s + 17/5·25-s + 4.31·26-s + 1.15·27-s − 2.26·28-s + 2.78·29-s − 4.38·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23833924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23833924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 2441 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 11 T + 55 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 15 T + 113 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 13 T + 3 p T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 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 | $D_{4}$ | \( 1 + 8 T + 77 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 + T + 91 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 13 T + 175 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 9 T + 151 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 5 T + T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 19 T + 217 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 13 T + 197 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - T + 177 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - T + 163 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−8.204513874868223814308970212935, −7.87297397150622926893476638345, −7.15237646138922532920434191508, −7.08173834065386388140076152182, −6.81796567415095488695590694805, −6.31320201893647900258288878633, −6.07841755836353710772232820179, −5.94040081171623001325274466711, −5.15784661485356643418222689040, −4.82891858328095370455616690684, −4.40849823732669966962110518506, −3.86690704719130534916813687200, −2.89398592018617880859825814018, −2.81949632790961834763436873999, −2.39071404726284683251771158695, −2.27858014172427954055809890869, −1.91320244254155060214020423346, −1.28438064210845473364995854712, 0, 0,
1.28438064210845473364995854712, 1.91320244254155060214020423346, 2.27858014172427954055809890869, 2.39071404726284683251771158695, 2.81949632790961834763436873999, 2.89398592018617880859825814018, 3.86690704719130534916813687200, 4.40849823732669966962110518506, 4.82891858328095370455616690684, 5.15784661485356643418222689040, 5.94040081171623001325274466711, 6.07841755836353710772232820179, 6.31320201893647900258288878633, 6.81796567415095488695590694805, 7.08173834065386388140076152182, 7.15237646138922532920434191508, 7.87297397150622926893476638345, 8.204513874868223814308970212935
