| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 2·5-s + 4·6-s + 2·7-s − 4·8-s + 3·9-s − 4·10-s − 6·12-s + 13-s − 4·14-s − 4·15-s + 5·16-s + 17-s − 6·18-s − 6·19-s + 6·20-s − 4·21-s + 5·23-s + 8·24-s + 3·25-s − 2·26-s − 4·27-s + 6·28-s − 29-s + 8·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s + 0.755·7-s − 1.41·8-s + 9-s − 1.26·10-s − 1.73·12-s + 0.277·13-s − 1.06·14-s − 1.03·15-s + 5/4·16-s + 0.242·17-s − 1.41·18-s − 1.37·19-s + 1.34·20-s − 0.872·21-s + 1.04·23-s + 1.63·24-s + 3/5·25-s − 0.392·26-s − 0.769·27-s + 1.13·28-s − 0.185·29-s + 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13176900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13176900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.472605675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.472605675\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - T + 25 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 42 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 5 T + 51 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + T + 27 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 51 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 69 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 10 T + 62 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 5 T + 81 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 25 T + 249 T^{2} - 25 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 7 T + 119 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 138 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 3 T + 75 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T - 34 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + T + 57 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 134 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 22 T + 270 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
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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
−8.731167500216219091023237826989, −8.665926877055264178417364411515, −7.82633832914397322403359288206, −7.68968460410172515156158838578, −7.17920260873558939203851114315, −7.12733170949391503150061085537, −6.38229205961320694963924402263, −6.17185905519244350333791108502, −5.87908044545276209384554790310, −5.62838606568684086816012750006, −4.90516484570415259197091493315, −4.80654937107170351980735626848, −4.11815855168949038905909172026, −3.80148436591959250304398652812, −2.88060757597995835958777114746, −2.59313387176211811118089270595, −1.85202457403829510734146406584, −1.75105267327218705344729360784, −0.792261685069056444484626206977, −0.71991040169048660568355329354,
0.71991040169048660568355329354, 0.792261685069056444484626206977, 1.75105267327218705344729360784, 1.85202457403829510734146406584, 2.59313387176211811118089270595, 2.88060757597995835958777114746, 3.80148436591959250304398652812, 4.11815855168949038905909172026, 4.80654937107170351980735626848, 4.90516484570415259197091493315, 5.62838606568684086816012750006, 5.87908044545276209384554790310, 6.17185905519244350333791108502, 6.38229205961320694963924402263, 7.12733170949391503150061085537, 7.17920260873558939203851114315, 7.68968460410172515156158838578, 7.82633832914397322403359288206, 8.665926877055264178417364411515, 8.731167500216219091023237826989
