| L(s) = 1 | − 2-s − 2·3-s − 2·4-s + 5-s + 2·6-s + 3·8-s − 10-s + 2·11-s + 4·12-s − 13-s − 2·15-s + 16-s + 6·17-s − 6·19-s − 2·20-s − 2·22-s − 2·23-s − 6·24-s − 25-s + 26-s + 2·27-s − 2·29-s + 2·30-s − 4·31-s − 2·32-s − 4·33-s − 6·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 4-s + 0.447·5-s + 0.816·6-s + 1.06·8-s − 0.316·10-s + 0.603·11-s + 1.15·12-s − 0.277·13-s − 0.516·15-s + 1/4·16-s + 1.45·17-s − 1.37·19-s − 0.447·20-s − 0.426·22-s − 0.417·23-s − 1.22·24-s − 1/5·25-s + 0.196·26-s + 0.384·27-s − 0.371·29-s + 0.365·30-s − 0.718·31-s − 0.353·32-s − 0.696·33-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 353 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 353 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1859131786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1859131786\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 353 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 9 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T - 8 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 75 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 142 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 92 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 74 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 14 T + 140 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 11 T + 116 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
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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
−19.9961442661, −19.3214919249, −18.7976671865, −18.2092074740, −17.6758825494, −17.1141440050, −16.8700603189, −16.3062469438, −15.0721190988, −14.3933411178, −13.8616163752, −12.9097577259, −12.3526386620, −11.5059400610, −10.7395302821, −9.94383563180, −9.33983492188, −8.54105304961, −7.57074114369, −6.16753924570, −5.48946407918, −4.23852001252,
4.23852001252, 5.48946407918, 6.16753924570, 7.57074114369, 8.54105304961, 9.33983492188, 9.94383563180, 10.7395302821, 11.5059400610, 12.3526386620, 12.9097577259, 13.8616163752, 14.3933411178, 15.0721190988, 16.3062469438, 16.8700603189, 17.1141440050, 17.6758825494, 18.2092074740, 18.7976671865, 19.3214919249, 19.9961442661
