| L(s) = 1 | − 2-s − 3·3-s − 4-s − 3·5-s + 3·6-s + 7-s + 8-s + 2·9-s + 3·10-s − 3·11-s + 3·12-s + 13-s − 14-s + 9·15-s + 3·16-s − 4·17-s − 2·18-s + 19-s + 3·20-s − 3·21-s + 3·22-s + 3·23-s − 3·24-s + 5·25-s − 26-s + 6·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 1/2·4-s − 1.34·5-s + 1.22·6-s + 0.377·7-s + 0.353·8-s + 2/3·9-s + 0.948·10-s − 0.904·11-s + 0.866·12-s + 0.277·13-s − 0.267·14-s + 2.32·15-s + 3/4·16-s − 0.970·17-s − 0.471·18-s + 0.229·19-s + 0.670·20-s − 0.654·21-s + 0.639·22-s + 0.625·23-s − 0.612·24-s + 25-s − 0.196·26-s + 1.15·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1706 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1706 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 853 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 8 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - 3 T - 13 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 52 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 14 T + 112 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T - 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 139 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + T + 102 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T - 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + T - 88 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 21 T + 237 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 170 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 7 T + 162 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
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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
−19.1218125615, −18.7415058911, −18.1237067731, −17.7341702303, −17.3864142015, −16.7552917508, −16.3523829886, −15.6907898780, −15.2601924823, −14.5656077421, −13.7141647510, −12.9783572525, −12.3207412923, −11.8380764703, −11.2823854458, −10.6983012846, −10.4932446269, −9.13724335851, −8.72764069983, −7.90022621316, −7.28954266110, −6.27210300810, −5.30435596953, −4.86647990260, −3.57535630494, 0,
3.57535630494, 4.86647990260, 5.30435596953, 6.27210300810, 7.28954266110, 7.90022621316, 8.72764069983, 9.13724335851, 10.4932446269, 10.6983012846, 11.2823854458, 11.8380764703, 12.3207412923, 12.9783572525, 13.7141647510, 14.5656077421, 15.2601924823, 15.6907898780, 16.3523829886, 16.7552917508, 17.3864142015, 17.7341702303, 18.1237067731, 18.7415058911, 19.1218125615
