| L(s) = 1 | − 3·2-s − 4·3-s + 4·4-s − 4·5-s + 12·6-s − 5·7-s − 3·8-s + 8·9-s + 12·10-s − 3·11-s − 16·12-s + 2·13-s + 15·14-s + 16·15-s + 3·16-s − 5·17-s − 24·18-s − 4·19-s − 16·20-s + 20·21-s + 9·22-s + 12·24-s + 6·25-s − 6·26-s − 12·27-s − 20·28-s − 48·30-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 2.30·3-s + 2·4-s − 1.78·5-s + 4.89·6-s − 1.88·7-s − 1.06·8-s + 8/3·9-s + 3.79·10-s − 0.904·11-s − 4.61·12-s + 0.554·13-s + 4.00·14-s + 4.13·15-s + 3/4·16-s − 1.21·17-s − 5.65·18-s − 0.917·19-s − 3.57·20-s + 4.36·21-s + 1.91·22-s + 2.44·24-s + 6/5·25-s − 1.17·26-s − 2.30·27-s − 3.77·28-s − 8.76·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3469 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3469 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3469 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 25 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 5 T + 15 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 31 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 11 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 5 T + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 18 T + 180 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 65 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 118 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 16 T + 135 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 77 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 88 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 131 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 3 T - 6 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
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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
−18.5433118016, −18.0132804163, −17.4854875395, −17.0712164772, −16.6292254044, −16.0795688583, −15.9105149703, −15.5888163926, −14.8196636355, −13.2631970348, −12.7877628891, −12.5193114341, −11.6157838344, −11.2796330892, −10.8548365891, −10.3553856332, −9.68857238955, −9.14493833937, −8.27148600182, −7.80313411940, −6.89888607101, −6.46359338194, −5.77120941456, −4.59854657271, −3.50911562665, 0, 0,
3.50911562665, 4.59854657271, 5.77120941456, 6.46359338194, 6.89888607101, 7.80313411940, 8.27148600182, 9.14493833937, 9.68857238955, 10.3553856332, 10.8548365891, 11.2796330892, 11.6157838344, 12.5193114341, 12.7877628891, 13.2631970348, 14.8196636355, 15.5888163926, 15.9105149703, 16.0795688583, 16.6292254044, 17.0712164772, 17.4854875395, 18.0132804163, 18.5433118016
